ASTRONOMY  DEPT. 


A    TREATISE 


ON 


PRACTICAL  ASTRONOMY, 


AS    APPLIED    TO 


GEODESY    AND    NAVIGATION. 


C    I  • 


BY 

C.    L.    DOOLITTLE, 
i  * 

Professor  of  Mathematics  and  Astronomy ',  Lehigh   University 
FOURTH  AND  REVISED  EDITION. 
THIRD     THOUSAND. 

NEW  YORK 
JOHN    WILEY    &   SONS 

LONDON 
CHAPMAN    &    HALL,    LTD. 

IQIO 


BY  C.  L.  DGOLITTLE. 

ASTRONOMY'' DEFT. 


PREFACE. 


THE  following  work  is  designed  as  a  text-book  for  univer- 
sities and  technical  schools,  and  as  a  manual  for  the  field 
astronomer.  The  author  has  not  sought  after  originality, 
but  has  attempted  to  present  in  a  systematic  form  the  most 
approved  methods  in  actual  use  at  the  present  time. 

Each  subject  is  developed  as  fully  as  the  necessities  of  the 
case  are  likely  to  require ;  but  as  the  work  is  designed  to 
be  a  practical  one,  those  methods  and  developments  which 
have  merely  a  theoretical  or  historic  interest  have  been  ex- 
cluded. 

Very  complete  numerical  examples  are  given  illustrative 
of  all  the  prominent  subjects  treated.  These  have  been 
selected  with  care  from  records  of  work  actually  performed, 
and  will  show  what  may  be  expected  in  circumstances  ordi- 
narily favorable. 

Such  auxiliary  tables  as  are  applicable  only  to  special  prob- 
lems will  be  found  in  the  body  of  the  work-  those  which 
have  a  wider  application  are  printed  at  the  end  of  the  volume. 

The  universal  employment  of  the  method  of  Least  Squares 
in  work  of  this  kind  has  led  to  the  publication  of  an  introduc- 
tion to  the  subject  for  the  benefit  of  those  readers  who  are 
not  already  familiar  with  it.  This  introduction  develops 
the  method  with  special  reference  to  the  requirements  of 


M30910 


IV  PREFACE. 

this  particular  class  of  work,  and  it  has  not  been  the  design 
to  make  it  exhaustive. 

For  the  materials  employed  original  papers  and  memoirs 
have  been  consulted  whenever  practicable.  The  illustrative 
examples  have  been  drawn  largely  from  the  reports  of  the 
Coast  and  other  government  surveys.  For  most  of  the  exam- 
ples of  sextant  work,  as  well  as  for  many  valuable  sugges- 
tions, the  author  is  indebted  to  his  friend  and  former  col- 
league Prof.  Lewis  Boss.  Much  assistance  has  also  been 
derived  from  the  excellent  works  of  Chauvenet,  Briinnow, 
and  Sawitsch. 

Fully  appreciating  the  difficulty  of  eliminating  all  mis- 
takes from  a  work  of  this  character,  the  author  can  only  hope 
that  this  one  may  not  prove  to  be  disfigured  by  an  undue 
number  of  such  blemishes. 

C.  L.  DOOLITTLE. 

BETHLEHEM,  PA.,  May  20,  1885. 


CONTENTS. 


INTRODUCTION  TO  THE  METHOD  OF  LEAST  SQUARES. 

PAGB 

Errors  to  which  observations  are  liable I 

Axioms. 2 

The  law  of  distribution  of  error 3 

The  curve  of  probability 5 

Determination  of  the  law  of  error 6 

Condition  of  maximum  probability u 

The  measure  of  precision 12 

The  probable  error 13 

The  mean  error 15 

The  mean  of  the  errors 17 

Precision  of  the  arithmetical  mean 18 

Determination  of  probable  error  of  arithmetical  mean 20 

Probable  error  of  the  sum  or  difference  of  two  or  more  quantities 22 

Principle  of  weights 23 

Probable  error  when  observations  have  different  weights 26 

Comparison  of  theory  with  observation 2g 

Indirect  observations 32 

Equations  of  condition — Normal  equations 35 

Observations  of  unequal  weight 36 

Arrangement  of  computation 37 

Computation  of  coefficients  by  a  table  of  squares 41 

Solution  of  normal  equations 43 

Proof-formulse 47 

Weights  and  probable  errors  of  the  unknown  quantities 54 

Mean  errors  of  the  unknown  quantities 65 


VI  CONTENTS. 


INTERPOLATION. 

PAGE 

Notation 71 

General  formulae  of  interpolation ' 72 

Arguments  near  beginning  of  table 78 

Arguments  near  end  of  table 82 

Interpolation  into  the  middle 84 

Proof  of  computation 85 

Differential  coefficients «, 86 

The  ephemeris — Lunar  distances *  92 


PRACTICAL   ASTRONOMY. 

CHAPTER   I. 
THE  CELESTIAL  SPHERE— TRANSFORMATION  OF  CO-ORDINATES. 

Spherical  co-ordinates 100 

The  horizon — Altitude — Azimuth 102 

The  equator— Declination — Hour- angle— Right  ascension 103 

The  ecliptic — Longitude — Latitude 104 

Having  altitude  and  azimuth,  to  find  declination  and  hour-angle 107 

Having  declination  and  hour-angle,  to  find  altitude  and  azimuth   112 

To  find  hour-angle  of  star  in  the  horizon 114 

To  find  distance  between  two  stars 115 

CHAPTER   II. 
PARALLAX — REFRACTION — DIP  OF  THE  HORIZON. 

Definitions 120 

To  find  equatorial  horizontal  parallax 120 

Parallax  at  any  zenith  distance 121 

Form  and  dimensions  of  the  earth 122 

Reduction  of  the  latitude 124 


CONTENTS.  Vll 

PAGE 

Determination  of  the  earth's  radius 127 

Parallax  in  zenith  distance  and  azimuth 131 

Parallax  in  right  ascension  and  declination 142 

Refraction *53 

Descartes'  laws 1 54 

Bessel's  formula  for  refraction 155 

Refraction  in  right  ascension  and  declination 157 

Dip  of  the  horizon 160 


CHAPTER   III. 

TIME. 

Sidereal  time 163 

Solar  time 164 

Inequality  of  solar  days 164 

Equation  of  time 166 

Sidereal  and  mean  solar  unit 168 

To  convert  mean  solar  into  sidereal  time 170 

To  convert  sidereal  into  mean  solar  time 172 


CHAPTER    IV. 

ANGULAR  MEASUREMENTS— THE  SEXTANT — THE  CHRONOMETER  AND 

CLOCK. 

The  vernier 174 

The  reading  microscope — The  micrometer 176 

Eccentricity  of  graduated  circles : . .  ,  180 

The  sextant 183 

The  prismatic  sextant 186 

Adjustments  of  the  sextant 188 

Method  of  observing 190 

Index  error 194 

Eccentricity  of  the  sextant 196 

The  chronometer 207 

Comparison  of  chronometers 208 

The  clock t . .  209 

The  chronograph 211 


Viii  CONTENTS. 

CHAPTER   V. 

DETERMINATION  OF  TIME  AND    LATITUDE — METHODS    ADAPTED    TO 
THE  USE  OF  THE  SEXTANT. 

PAGE 

Determination  of  time — 

By  a  single  altitude  of  the  sun 215 

By  a  single  altitude  of  a  star i 220 

Conditions  favorable  to  accuracy 222 

Differential  formulae 223 

Equal  altitudes  of  a  star 228 

Equal  altitudes  of  the  sun 230 

Latitude  233 

By  the  zenith  distance  of  a  star  on  the  meridian 233 

By  a  circumpolar  star  observed  at  both  upper  and  lower  culmination. .   235 

By  the  altitude  of  a  star  observed  in  any  position 236 

By  circummeridian  altitudes 238 

Gauss'  method  of  reducing  circummeridian  altitudes  of  the  sun  247 

Correction  for  rate  of  chronometer 250 

Latitude  by  Polaris 256 

Correction  of  altitudes  for  second  differences  in  time 260 

Probable  error  of  sextant  observation 265 

CHAPTER   VI. 
THE  TRANSIT  INSTRUMENT. 

Description  of  instrument 269 

Value  of  level f '. . . . .  276 

Adjustments  of  instrument 279 

Methods  of  observing . . . .  283 

Theory  of  the  transit 284 

Diurnal  aberration 289 

Equatorial  intervals  of  threads 291 

Reduction  of  imperfect  transits 294 

Determination  of  constants: 

The  level  constant 295 

Inequality  of  pivots 296 

The  collimation  constant 302 

The  azimuth  constant 305 

Personal  equation 316 

Probable  error  and  weight  of  transit  observations 318 

Application  of  the  method  of  least  squares 322 


CONTENTS.  IX 

PAGE 

Correction  for  flexure 335 

The  transit  instrument  out  of  the  meridian 338 

Transits  of  the  sun,  moon,  and  planets 339 

Correction  to  moon's  defective  limb 343 

The  transit  instrument  in  the  prime  vertical 348 

Mathematical  theory 352 

Errors  in  the  data 356 

Reduction  to  middle  or  mean  thread 358 

Application  of  least  squares  to  prime  vertical  transits 372 


CHAPTER   VII. 
DETERMINATION  OF  LONGITUDE. 

By  transportation  of  chronometers 379 

By  the  electric  telegraph 388 

By  the  moon .- 398 

By  lunar  distances 400 

By  moon  culminations   413 

By  occultations  of  stars 423 

Prediction  of  an  occultation 435 

Graphic  process  of  prediction 443 

Computation  of  longitude 444 

Correction  for  refraction  and  elevation  above  sea-level 460 

Observations  of  different  weights 474 

CHAPTER   VIII. 

THE  ZENITH  TELESCOPE. 

Description  of  instrument 478 

Adjustment  of  instrument  481 

The  observing  list 484 

Directions  for  observing 48" 

Value  of  micrometer  screw 488 

Value  of  micrometer  when  level  is  not  known 493 

General  formulae  for  latitude 5°* 

The  corrections  for  micrometer,  level,  and  refraction 5°2 

Reduction  to  the  meridian 5°4 

Combination  of  individual  values  of  the  latitude ; 5°7 

Value  of  micrometer  from  latitude  observations 5°9 


X  CONTENTS. 

CHAPTER   IX. 
DETERMINATION  OF  AZIMUTH. 

PAGE 

The  theodolite 521 

The  signal 523 

Selection  of  stars — Method  of  observing 524 

Errors  of  collimation  and  leVel 525 

Azimuth  by  a  circumpolar  star  near  elongation 526 

Correction  for  diurnal  aberration 530 

Circumpolar  stars  at  any  hour-angle 535 

Correction  for  second  differences  in  the  time 537 

Conditions  favorable  to  accuracy , 542 

Azimuth  when  time  is  unknown 543 

Azimuth  determined  by  transit  instrument 546 

Circumpolar  star  at  any  hour-angle 552 


CHAPTER  X. 
PRECESSION — NUTATION — ABERRATION — PROPER  MOTION. 

Secular  and  periodic  changes 559 

Mean,  apparent,  and  true  place  of  a  star. 560 

Precession 560 

Struve  and  Peters'  constants 563 

Bessel  and  Leverrier's  constants 564 

Precession  in  longitude  and  latitude 564 

Precession  in  right  ascension  and  declination 571 

Proper  motion 57§ 

Expansion  into  series 5§3 

Star  catalogues  and  mean  places  of  stars 590 

Nutation 59§ 

Aberration 603 

Reduction  to  apparent  place 609 

The  fictitious  year 617 

The  Tabula  Regiomontance 620 

Conversion  of  mean  solar  into  sidereal  time 623 

LIST  OF  TABLES 626 


INTRODUCTION   TO   THE   METHOD    OF 
LEAST  SQUARES. 


1.  When  a  quantity  is  determined  by  observation,  the  re- 
sult can  never  be  regarded  otherwise  than  as  an  approxima- 
tion to  the  true  value.     If  a  number  of  measurements  of  the 
same  quantity  are  made  with  extreme  care,  no  two  of  the 
values  obtained  will  probably  agree  exactly ;  at  the  same 
time  none  of  them  will  differ  very  widely  from  the  true  one. 

There  is  a  limit  to  the  precision  of  the  most  refined  instru- 
ment, even  when  usecl  by  the  most  skilful  observer,  and 
therefore  the  determination  of  a  quantity  depending  on  in- 
strumental measurement,  however  carefully  made,  must  be 
imperfect.  It  becomes  then  a  problem  of  great  practical 
importance  to  determine  how  the  mass  of  data  resulting  from 
observation  shall  be  combined  so  as  to  give  the  best  possible 
value  of  the  quantity  sought.  The  theory  of  probabilities 
furnishes  the  basis  for  such  an  investigation.* 

2.  Observations  are  liable  to  errors  of  three  kinds : 
First.  Constant  errors,  or  those  which  affect  all  observa- 

*  The  reader  is  supposed  to  be  familiar  with  the  theory  of  probability  as  de- 
veloped in  the  ordinary  text-books  on  algebra.  See,  for  instance,  Davies 
Bourdon,  edition^of  1874,  p.  322,  or  Olney's  University  Algebra,  p.  294. 


2  LEAST  SQUARES.  §  3. 

tions  of  a  given  series  alike.  These  may  result  from  a 
variety  of  causes,  such  as  errors  in  the  instruments  used, 
personal  error  of  the  observer,  errors  in  the  constants  of  re- 
fraction, parallax,  etc.,  used  in  the  reduction  of  observations. 
A  proper  investigation  will  generally  show  the  magnitude  of 
such  errors,  and  consequently  the  necessary  corrections — at 
least  the  more  important  ones.  We  shall  suppose  the  data 
to  which  our  discussion  applies  freed  from  such  errors,  as 
their  investigation  does  not  come  within  the  scope  of  this 
subject. 

Second.  Mistakes,  such  as  recording  the  wrong  degree  in 
measuring  an  angle,  or  the  wrong  hour  in  the  clock  reading. 
When  such  errors  are  large  they  are  not  likely  to  give  much 
trouble,  as  their  true  nature  appears  at  once.  When  they  are 
small  they  may  prove  embarrassing.  The  present  discussion 
does  not  apply  to  them,  and  we  shall  suppose  that  no  undis- 
covered mistakes  have  been  made. 

Third.  Errors  which  are  purely  accidental.  It  is  to  these 
that  our  present  investigation  applies. 

At  first  sight  it  might  seem  that  such  purely  accidental 
errors  were  entirely  outside  the  sphere  of  mathematical  in- 
vestigation, but  we  shall  see  that  they  follow  a  very  definite 
law,  and  that  theory  is  verified  in  an  exceedingly  satisfactory 
manner  by  observation. 

3.  We  shall  assume  as  the  basis  of  our  investigation  the 
following  axioms : 

I.  If  we  have  a  series  of  direct  measurements  of  a  quantity, 
all  made  with  equal  care,  the  most  probable  value  of  the 
quantity  will  be  obtained  by  taking  the  arithmetical 
mean  of  the  individual  measurements. 
II.  Plus  and  minus  errors  will  occur  with  equal  frequency. 
III.  Small  errors   will  occur   with  greater  frequency   than 
large  ones.  + 


§  4-  DISTRIBUTION   OF  ERRORS.  3 

Various  attempts  have  been  made  to  prove  the  first  of 
these  as  a  proposition.  All  such  proofs  are  more  or  less 
unsatisfactory,  and  for  elementary  purposes  it  is  more  ex- 
pedient to  assume  its  truth  at  once.  The  "  most  probable 
value"  there  mentioned  must  be  understood  as  the  value 
which  most  nearly  represents  the  given  data,  and  from  the 
evidence  furnished  by  this  series  of  observations  alone  it  is 
the  best  attainable  approximation  to  the  true  value. 

The  principles  are  supposed  in  all  cases  to  be  applied  to  a 
large  number  of  observations;  the  larger  the  number  the 
more  closely  will  the  results  correspond  to  the  laws  assumed. 


The  Law  of  Distribution  of  Error. 

4.  Let  x  be  a  quantity  whose  value  is  to  be  determined 
by  observation  either  directly  or  indirectly. 

Let  MV  M»  MV  .  .  .  Mm  be  the  individual  values  obtained. 

Then  regarding  Ml  as  a  determination  of  the  unknown 
quantity  x,  its  error  will  be  (Ml  —  x).  Similarly,  (Ma  —  x], 
(Mz  —  x),  .  .  .  (Mm  —  x)  will  be  the  errors  of  the  other  ob- 
served values. 

Let  us  write 

(M,  ~  *)  =  A,  (M*  -*)  =  *„...  (Mm  -*)=  Am.    (i) 

Let     y,  —  the  probability  of  the  occurrence  of  the  error  Al ; 
y^  =  the  probability  of  the  occurrence  of  the  error  A^  • 


ym  —  the  probability  of  the  occurrence  of  the  error  Am. 

Then  our  second  and  third  axioms  assume  a  law  as  existing 
such  that  the  probability  of  a  given  error  occurring  will  be 


4  LEAST  SQUARES.  §  4. 

a  function  of  the  magnitude  of  the  error  itself.     We  shall 
therefore  have  the  equation 


y  =  ¥^\  ........   (2) 

in  which  A  represents  any  error,  and  y  the  probability  of  its 
occurring. 

If  this  reasoning  seems  obscure,  a  different  application  of 
the  same  logic  may  possibly  assist  in  comprehending  it. 
Suppose  we  have  a  large  number  of  tickets  in  a  lottery- 
wheel.  Let  a  definite  proportion  of  them  be  numbered  i, 
a  certain  other  proportion  respectively  2,  3,  etc.  Then 
the  probability  of  drawing  any  given  number  from  the  wheel 
will  be  a  function  of  the  number  itself  —  viz.:  ^ 

Suppose       I  ticket    in  every  55  numbered  I 

2  tickets  in  every  55  numbered  2 

3  tickets  in  every  55  numbered  3 


10  tickets  in  every  55  numbered  10. 

0 

Then  every  ticket  would  have  one  of  the  numbers,  i,  2,  3,  4, 
5.  6>  7»  8,  9,  10,  and 

The  probability  of  drawing  a  i  would  be  -gijj 
The  probability  of  drawing  a  2  would  be  -/%; 
The  probability  of  drawing  a  10  would  be  ££. 

Or  if  k  represents  any  one  of  the  numbers  from  i  to  10  in- 
clusive, the  probability  of  drawing  a  k  will  be  —   =/(&),  or 

y  =-J\k)  is  the  equation  which  represents  the  probability  of 
drawing  a  k. 

If  now  we  were  ignorant  of  the  relations  existing  between 
the  successive  numbers  I,  2,  3,  etc.,  and  the  relative  number 


§5- 


CURVE    OF  PROBABILITY. 


of  tickets  so  marked,  we  could,  by  drawing  a  sufficiently  large 
number  of  tickets  from  the  wheel,  determine  it,  at  least  ap- 
proximately. In  this  case  we  have  to  determine  the  proba- 
bility of  a  given  event  occurring,  viz.,  that  of  drawing  a 
ticket  marked  with  any  given  number  k.  In  the  above  prob- 
lem we  have  also  to  discuss  the  probability  of  a  certain  event 
occurring,  viz.,  that  of  the  appearance  of  any  given  error  A 
in  any  one  of  our  observations  taken  at  random. 

The  Curve  of  Probability. 

5.  In  the  equation  y  =  *<p(A\  we  can  regard  A  as  the  ab- 
scissa, and  y  as  the  ordinate  of  a  curve.  From  the  laws  pre- 
viously assumed  we  at  once  infer  that  the  general  form  of 
the  curve  will  be  that  of  the  following  figure.  In  the  first 
place,  as  +  and  --  errors  are  equally  probable,  it  follows 
that  the  curve  will  be  symmetrical  with  respect  to  the  axis 
of  y ;  and  as  small  errors  are  more  probable  than  large  ones, 
it  follows  that  the  values  of  A  near  zero  will  correspond  to 
large  values  of  j/,  while  as  A  becomes  very  large  y  becomes 
very  small. 

Y 


M     P 


P     M 


FIG.  i. 


Practically  A  is  not  a  continuous  variable,  and  our  locus, 
therefore,  consists  of  a  series  of  disconnected  points.     The 


6  LEAST  SQUARES.  §  6. 

intervals  between  the  different  values  of  A  will  DC  equal  to 
the  smallest  reading  of  the  instrument  with  which  the  obser- 
vations were  made.  The  greater  the  degree  of  precision  in 
the  data,  however,  the  more  closely  will  our  locus  approach 
continuity  ;  so  by  regarding  it  as  a  continuous  curve  we  have 
a  condition  towards  which  we  are  constantly  approximating 
as  methods  of  observation  become  more  and  more  refined. 


Determination  of  the  Function  cp. 
6.  For  the  probability  of  an  error  A  we  have  the  equation 

y  =  <?(/0; 

and  for  an  error  A  -\-  6Ay 

y'  =  <p(A  + 


The  probability  that  an  error  falls  between  A  and  A  -f-  d  A 
will  be  the  sum  of  all  the  probabilities  between  y  and  y'\  or 
if  6  A  is  small,  it  will  be  nearly  dAcp(A).  When  d  A  becomes 
dA,  we  have  rigorously  y  =  cp(A)dA  for  the  probability  that 
an  error  falls  between  A  and  A  -|-  dA*  For  the  probability 
of  an  error  falling  between  any  finite  limits,  as  for  instance 

*  By  way  of  illustration  let  us  suppose  the  smallest  unit  of  measure  made 
use  of  in  our  observations  tobeo".i,  and  that  any  given  number  of  these  units, 
as  for  instance  3,  are  represented  by  d  A.  Then  the  errors  between  A  and 
A  -|-  8  A,  including  the  latter,  will  be  (A  -f-  i),  (A  -f-  2),  and  (A  -j-  3);  and  their 
respective  probabilities,  jpi  =  <p(A  -\-  i),  y*  —  cp(A  -f-  2),  and  ya  =  (p(A  +  3). 
If  now  the  limits  between  which  the  errors  of  our  series  lie  extend  to  ±  10", 
we  see  that  the  probability  y\  will  differ  but  little  from^3,  and  the  sum  of  all  the 
probabilities  y*  -\-y*-\-ys  will  differ  but  little  from  3^,  or 

8  Ay  =  q>(A)SA. 


§6.  DETERMINATION   OF    THE   LAW  OF  ERRORS.  *J 

±  a,  we  shall  have  the  sum  of  the  probabilities  for  all  values 
of  A  between  ±  a,  or 


(3) 


When  we  extend  the  limits  of  integration  so  as  to  include 
all  possible  values  of  A,  the  probability  becomes  a  certainty, 
which  is  expressed  mathematically  by  unity.  As,  however, 
it  is  impossible  to  fix  a  finite  limit  to  the  value  of  A  which 
shall  be  universal  in  its  application,  the  limits  in  this  case 
must  be'extended  to  ±  oo,  giving-  us  the  eguation 


From  the  foregoing  we  have 


for  the  probability  of  the  error 
2  =  <p(A^)  for  the  probability  of  the  error 


ym  =  <p(Am)  for  the  probability  of  the  error  Am. 

If  now  P=  the  probability  that  all  these  errors  occur  si- 
multaneously, we  have,  from  the  theory  of  probabilities, 

P=  ?(4)<K4MA)    -    -    •  9(4.),    ...    (5) 

and  the  most  probable  value  of  the  unknown  quantity  x  will 
be  that  which  makes  the  quantity  Pa.  maximum. 

Taking  the  logarithms  of  both  members  of  this  equation, 
we  have 

logP  =  log  <p(A)  +  log  9<4)  +  .  .  .  +  log  <?(4n). 


8  LEAST  SQUARES.  §  6. 

Differentiating  this  with  respect  to  x>  and  placing  the  dif- 
ferential coefficient  equal  to  zero,  which  is  the  condition  of  a 
maximum,  we  have 


, 

<fcr  dA  dx 


From  (i)  we  have 


dx     ~  dx  ~  dx 


Substituting  these  values  in  the  above  equation,  also  for 
etc.,  their  values  (M,  —  x),  etc.,  it  becomes 

d  log  (p(M,  -  x]       d  log  cp(M,  -  x) 
d(M,  -  x)  d(M,  -  x) 


This  equation  gives  the  means  of  determining  x  as  soon 
as  the  form  of  the  function  cp  is  known,  and  this  can  best  be 
determined  by  considering  a  particular  case.  As  this  func- 
tion is  strictly  general,  if  we  have  once  determined  its  form 
in  a  special  case  the  result  will  be  applicable  to  all  cases. 

We  have  assumed  as  an  axiom  that  in  the  case  of  direct 
measurement  of  the  quantity  sought  the  most  probable  value 
will  be  the  arithmetical  mean  of  the  individual  measurements. 
This  principle  will  furnish  the  basis  for  investigating  the 
form  of  the  function  cp. 

In  case  of  direct  measurement  we  have  for  the  unknown 
quantity 

Mt  +  M,+  ...+Mm 
*--  ~  -,...-    (7) 


§6.  DETERMINATION  OF   THE  LAW  OF  ERRORS.  9 

which  may  be  written 

(Ml-x)  +  (M,-x)  +  ...+(Mm-x)  =  o.     .    (8) 
Equation  (6)  may  be  written 


- 


Comparing  equations  (3)  and  (9),  we  see  that  since  the 
quantities  (Ml  —  x),  (Mz  —  x\  etc.,  are  independent  of  each 
other,  these  equations  may  be  satisfied  by  placing  the  coeffi- 
cients of  (Ml  —  x),  (Mt  —  x],  etc.,  in  (9),  respectively  equal 
to  the  same  .constant,  k.  We  have  therefore 

dlog  (?(]$,—  x)  d\og(p(M^  —  x) 

(M,  -  x]d(M,  -x)~  (Mt  -  x}d(M^  -  x) 

-x) 

- 


Writing  for  (M  —  x)  in  general  //,  we  have 


and,  by  integration,     log  (p(A)  =  \kA*  +  log  c, 

c  being  the  constant  of  integration, 

or  <p(4).=  ce**  .....    .     .'.     (ii) 


From  axiom  III.  it  appears  that  as  A  increases  this  quan- 
tity must  diminish,  and  this  requires  the  exponent  of  e  to  be 


10  LEAST  SQUARES.  §  7. 

negative.     As  J2  cannot  be  negative,  it  follows  that  k  must 
be  so.     Writing  therefore  \k  =  —  tf,  our  equation  becomes 

ce-**.     I    .....     (12) 


7.  Let  us  now  consider  the  constant  of  integration  c.  This 
may  be  determined  by  substituting  the  value  of  <p(d)  in  (4), 
giving  us 


a  special  form  of  the  integral  known  as  the  gamma  function. 
For  the  purpose  of  integrating  the  expression,  place  /id  =  t. 

Then  dA  =  ->-,  and  we  have 

/+«>/•  r    /*+  °° 

CTe~*dt=  rj        e-»dt. 
oo      Jt  h^-" 

As  /  in  this  expression  is  involved  only  in  the  quadratic 
form,  we  evidently  have 


(in  which  we  write  the  integral  equal  to  A  for  convenience). 

/oo 
e~*dt  the  value  will  be  the  same 

if  we  write  another  symbol  instead  of  t.     Therefore 
.     C  e~*dt  =  f  e~*dv. 

I/O  t/O 

e-*dt,  we 
have 

A'=  f    f  e- 

t/o       t/o 


§  8.  CONDITION   OF  MAXIMUM  PROBABILITY.  1 1 

In   the   second    member  of  this  equation  write  v  =   tu, 
dv  —  tdu.     Then 

A*=  TduTe- 

I/O  t/0 


/ 
'- 


which  between  the  given  limits  becomes  +    ,  i-. 

Therefore 

A"  =  -  I    —  ;  —  5  =  -(tan-  x  oo  —  tan-  x  o)  =  —it. 

2  e/o       I  -[_  U*          2  ^  }         4 


Therefore 


and  we  have  i  =  j  J/TT,        or 

r    **• 

and  equation  (12)  becomes 

B— ^*.*W (I3) 


In  this  equation  the  constant  h  will  require  further  con- 
sideration ;  but  if  we  assign  any  arbitrary  value,  as  unity,  to 
h  we  can  readily  construct  the  locus  of  the  equation.  It  will 
at  once  appear  that  the  general  form  will  be  that  shown  on 
page  5. 

/ 

Condition  of  Maximum  Probability  ', 

8.  Substituting  in  equation    5)  the  values  of 
etc.,  from  (13),  it  becomes 


P—  /-_\-/ia(^2+A22  +  ---+^a).     .    .     .    (14) 


12  LEAST  SQUARES.  §9. 

From  this  equation  we  see  that  P  will  increase  in  value  as 
the  exponent  of  e  diminishes,  or  P  will  be  a  maximum  when 
4?  +  d*  -\-  .  .  .  +  A^  is  a  minimum,  thus  giving  us  the  im- 
portant principle— 

The  most  probable  value  of  the  unknown  quantity  is  that  zvhich 
'makes  the  sum  of  the  squares  of  the  residual  errors  a  minimum. 

From  this  principle  comes  the  name  Method  of  Least  Squares. 


The  Measure  of  Precision. 

9.  Let  us  now  consider  the  constant  h. 
Substituting  in  equation  (3)  the  value  of  <p(A\  we  have  for 
the  probability  of  an  error  between  the  values  ±  a 

/+<*    h 
,  ~*-Wd4 (15) 

If  we  take  another  series  of   observations,  we  have  the 
probability  of  an  error  between  ±  af 


If  these  respective  probabilities  are  equal  we  shall  have 


which  equation  will  be  satisfied  by  making  ha  =.h'af,  or 

h  :  h'  =  a'  :  a  .......     (16) 

We  see  from  this  equation  that  in  two  different  series  of 
observations  h  will  have  different  values,  these  values  being 


§  10.  THE   PROBABLE   ERROR.  13 

to  each  other  inversely  as  the  errors  to  be  ascribed  with  equal 
probability  to  each  series.  If,  for  instance,  the  errors  of  the 
first  series  are  twice  as  great  as  those  of  the  second,  h  will 
equal  \h' .  The  constant  h  is  therefore  the  measure  of  pre- 
cision of  the  series  of  observations ;  and  if  its  value  could  be 
determined  from  the  observations  themselves,  we  should  by 
this  means  be  able  to  know  to  what  degree  of  confidence  the 
data  were  entitled.  This  determination  is  possible, — at  least 
approximately, — but  for  practical  purposes  it  is  more  con- 
venient to  compare  the  relative  accuracy  of  different  series  of 
observations  by  means  of  their  respective  probable  errors, 
which  will  now  be  considered. 


The  Probable  Error. 

10.  The  probable  error  of  any  observation  of  a  given  series 
is  a  quantity  such  that  if  the  errors  committed  be  arranged 
according  to  their  magnitude  without  reference  to  the 
algebraic  sign,  this  quantity  will  occupy  the  middle  place  in 
the  series.  //  may  therefore  be  defined  as  a  quantity  of  such 
value  that  the  probability  of  an  error  greater  than  this  one  is  the 
same  as  the  probability  of  one  less. 

When  we  consider  both  plus  and  minus  errors,  we  have 
from  equation  (i  5)  the  following  expression  for  the  probability 
of  an  error  between  ±  a,  remembering  that  the  probability 
between  o  and  +  a  is  the  same  as  between  o  and  —  a  : 


Let  r  =  the  probable  error. 

The  whole  number  of  errors  being  represented  by  unity, 


14  LEAST  SQUARES.  §  II. 

our  definition  of  the  probable  error  gives  us  the  following 
equation : 

l-=^Te- »"M,        or        -  =  -^S*' - ™hdA.  ( 1 8) 

2  4/7T^°  ^X 

The  solution  of  this  equation  will  give  us  hr ;  so  that  if  h  is 
known  r  becomes  known,  and  conversely. 

II.  It  is  evident  that  the  equation  for  hr  can  only  be 
solved  approximately,  as  the  expression  e-^^lidA  is  not 
directly  integrable.  The  only  method  of  solution  is  to  com- 
pute a  series  of  numerical  values  of  the  integral  for  different 
values  of  the  limit,  hr,  and  then  by  interpolation  determine 
that  value  whichr  satisfies  equation  (18)  with  the  necessary 
degree  of  precision. 

Owing  to  the  great  importance  of  this  integral,  not  only 
in  this  connection,  but  also  in  the  theory  of  refraction,  vari- 
ous methods  have  been  developed  for  computing  its  numeri- 
cal value.  The  most  elementary  of  these  consists  in  expand- 
ing e~h<i^  =  e~  *  (hd  being  written  equal  to  /)into  a  series  ot 
ascending  powers  of  t,  by  means  of  Maclaurin's  formula,  and 
integrating  the  separate  terms  of  the  series.  This  series 
converges  rapidly  for  small  values  of  /,  and  is  therefore  well 
adapted  to  numerical  computation,  but  for  large  values  of  t 
it  becomes  diverging.  For  this  case,  as  well  as  for  the  case 
where  /  is  small,  a  series  may  be  obtained  by  successive  ap- 
plications of  the  formula  for  integration  by  parts, 


J  udv  —  uv  —J 


vdu, 

by  which  means  the  expansion  may  be  effected  either  in 
terms  of  ascending  or  descending  powers  of  /.  When  an 
extensive  series  of  values  of  the  integral  is  required,  as  in 
computing  a  table  of  values  for  different  values  of  the  argu- 


§  12.  THE   MEAN  ERROR.  1 5 

ment,  /,  the  most  simple  process  is  to  apply  what  is  known 
as  the  method  of  Mechanical  Quadratures. 

As  very  complete  tables  of  numerical  values  of  this  integral 
have  been  many  times  computed,  we  shall  simply  refer  to  the 
tabular  quantities  without  entering  more  fully  into  the  methods 
of  computation.  Table  I.  of  this  volume  gives  the  values  of 

-   I  e-*  dt  for  values  of  t  from  o  to  oo .     We  readily  find 

V  X   J  o 

from  this  table  that  the  value  of  hr  which  satisfies  equation 
(18)  lies  between  .47  and  .48.     An  interpolation  readily  gives 

hr  —  0.47694; 
.47694 


47^94 


_ 

h 


The  Mean  Error. 

12.  The  probable  error  is  not  the  only  function  of  the 
errors  which  may  be  used  for  comparing  the  relative  ac- 
curacy of  different  series  of  observations.  Another  quantity 
which  may  be  used  for  this  purpose,  or  as  a  convenient  aux- 
iliary for  computing  the  probable  error,  is  the  Mean  Error. 

The  Mean  Error  is  a  quantity  whose  square  is  the  mean  of 
the  squares  of  the  individual  errors. 

Let  £  =  the  mean  error.  Then  to  determine  the  relation 
between  f  and  h,  and  consequently  between  e  and  r,  we  pro- 
ceed as  follows :  Let 

A' ,       An ',       A'" ,    etc.  =  the  different  errors  which  occur  ; 
'\  (p(A"\  (p(A"r),  etc.  =  their  respective  probabilities. 


l6  LEAS7"  SQUARES.  §  12. 

Then  m  being  the  whole  number  of  errors5  there  will  be  a 
number  expressed  by  the  quantity  2mq)(A')  (both  -f-  and  — 
errors  included)  of  the  value  A'  ,  2mcp(A")  of  the  value  A1', 
etc.,  and  in  all 

4  +  4,  +  A,  +  .  .  .  Am  =  2mcp(A')Af  +  2m<p(A")A" 

4-  2^?>(^//  V"  +  etc. 


From  the  definition  of  the  mean  error  £  we  shall  have 


2mcp(A')A''i  +  2m9(A"}A'">  +  2^(J///)^///2  +  etc. 


m 


Expressing  this  by  an  integral,  by  the  same  method  of  rea- 
soning as  was  used  in  deriving  equation  (3)  we  have 


/CO          If 
~ 


This  equation  expresses  a  relation  between  e  and  h.  To 
effect  the  integration,  let  as  before  hA  —  t.  Then  dA  =  -^-, 
and  we  have 


Integrating  this  by  parts  by  placing  u  —  t  and  dv  =  e  -  **tdt, 
and  substituting  in  J  udv  =  uv  —J  vdu,  we  find 


which  readily  gives  £a  —  -     .........     (20) 


§13-  THE  MEAN  OF   THE   ERRORS.  I? 

Substituting  the  value  of  h  from  (19),  we  have 


">  \    .  (21) 

r=     .6745*.) 

From  these  r  is  readily  computed  when  we  know  e,  and  vice 
versa. 


The  Mean  of  the  Errors. 

13.  Another  quantity  which  is  much  used  as  an  auxiliary 
for  computing  r  is  The  Mean  of  the  Errors.  This  must  not 
be  confused  with  the  mean  error.  It  is  thus  defined  : 

The  Mean  of  the  Errors  is  the  arithmetical  mean  of  the  differ- 
ent errors  all  taken  with  the  positive  sign. 

Let  rj  =  the  mean  of  the  errors.  Then  to  determine  the 
relation  between  rj  and  r  we  proceed  in  a  manner  similar  to 
that  followed  in  the  previous  section.  As  before,  let 

A'  ,        A"  ,        A'n  ',    etc.  =  the  individual  errors. 
<p(A'\    (p(A"},    <p(A'"},   etc.  =  their  respective  probabilities. 

Then,  the  whole  number  of  observations  being  m, 

mrj  =  2m  <p(A')A'  +  2m  <p(A"}A"  '+  2m  <p(A'")A"r,  etc., 

from  definition  ;  and  therefore 

7;  =  2m<p(A'}  4'+2m<p(4")4"+2m<p(4"y",  etc.  =  2^(J)J> 

Passing  to  the  integral  as  before,  m  being  supposed  very 
large, 

"  =  -^=.      .     .     .     (22) 

h  V* 


18  LEAST  SQUARES.  §  14. 

Substituting  the  value  of  h  from  (19), 

r,  =  I.i829r;  ) 
r  =  0.8453'A  f 

Equations  (20)  and  (22)  give  us  the  following  relations  be- 
tween e  and  ?/,  which  we  shall  hereafter  find  convenient : 


(24) 


Either  of  the  quantities  r,  £,  or  17  may  be  used  for  comparing 
the  relative  accuracy  of  different  series  of  observations,  or  of 
the  quantities  derived  from  them  by  computation.  We  shall, 
however,  always  use  r  for  this  purpose,  making  use  of  rj  and 
£,  when  occasion  serves,  as  convenient  auxiliaries  for  comput- 
ing the  probable  error  r. 


Precision  of  the  Arithmetical  Mean. 

14.  Although  the  arithmetical  mean  is  the  best  value  to  be 
obtained  from  a  series  of  equally  good  direct  measurements, 
it  will  only  be  an  approximation  to  the  true  value.  It  is  there- 
fore important  to  determine  to  what  degree  of  confidence  it 
is  entitled.  Let 

»„  n»  nz, . .  .nm  =  m  individual  measurements  of  the  quantity  x\ 
A^AVA^  . .  .Am  =  the  errors,  of  each  n  respectively. 

Then      **=(*,-*  A)  =  (*,  -  O  =  •  •  •  =  (nm  -  40- 


§14.  THE  ARITHMETICAL   MEAN.  19 

Then  the  most  probable  value  of  x  is  that  which  gives  a 
maximum  value  to  the  expression 
h 


=  ( 

V 


Vn 

A*  -\-  A*  -|-  .  .  .  -\- ^m  must  therefore  be  a  minimum. 
But  this  is  the  case  when  x  is  the  arithmetical  mean  of  the 
individual  values  «,,  ;;.,,  .  .  .  nm. 

If  any  other  value  is  assumed  for  x,  as  x  -f-  #,  the  cor- 
responding errors  will  be  ^,  —  tf,  J2  —  #,...  Jm  —  S,  and 
(14)  becomes 

/  h    \m 

P!  (_        _)  _^2[(^l_i 

~\Vx> 

But  from  article  12,      24'  —  ;^e3.      Also,      ^A  =  o, 
Therefore  (14)  and  (I4)1  become 


if 

or  />:/>/=  i:*-"*1" (25) 

For  a  single  observation  m  =  i,  and  this  expression 
becomes 

/*:>/=*  i:«r^ (25), 

Therefore  //  being  the  measure  of  precision  of  the  indi- 
vidual observations,  that  of  the  arithmetical  mean,  of  m  such 

observations  is  h  Vm. 

Therefore,  calling  h,  the  measure  of  precision,  e0  and  r0  the 
mean  and  probable  errors  of  the  arithmetical  mean,  we  have, 
from  formulae  (19),  (21),  (25),  and  (25),, 


20  LEAST   SQUARES.  §  15. 

That  is,  the  precision  of  a  result  obtained  by  direct  measurement 
is  directly  as  the  square  root  of  the  number  of  measurements. 


Determination  of  the  Probable  Error. 

15.  From  the  foregoing  principles  we  can  now  compute 
from  the  observations  themselves  the  probable  error  of  a 
quantity  determined  directly  by  observation. 

As  before,  let  n^  n^  ns,  .  .  .  nm  —  the  individual  measure- 
ments of  a  quantity  x. 

Let        x0  =  the  arithmetical  mean  of  the  n's  ; 


These  quantities  (vlt  v9t  etc.)  are  known  as  residuals,  and 
must  not  be  confounded  with  the  true  errors  (A^  Av  etc.), 
from  which  they  will  always  differ,  unless  x^  is  absolutely  the 
true  value  of  x. 

Let  the  error  of  x,  be  6.     Then  x  =  x0  -\-  tf,  and  conse- 
quently 


and  we  shall  have 

[J  J]  r=  [W]  -  2\v\8  +  m&  ; 

in  which  \yv\  *  =  v*  +  v*  +  .  .  .  +  z>m2 

and  [>]  *    =  ^   +  va   +  .  .  .  -|~  vm. 

Since  jr0  is  the  arithmetical  mean  of  the  quantities  nlt  «a, 
etc.,  it  follows  that  \v\  =  o,  and  consequently 


*  Frequent  use  will  be  made  hereafter  of  this  symbol  of  summation,  and  it 
will  require  no  further  explanation. 


DETERMINATION   OF  PROBABLE  ERROR. 


21 


d  being  the  error  of  the  arithmetical  mean,  is  unknown.  A 
close  approximation  will,  however,  be  obtained  if  we  assume 
it  equal  to  the  mean  error  e0*  Then  referring  to  (25),  we  have 


—  m —  — 
1    m 


and  since  \_44~\  =  mf,  we  have 
m?  =  \yv\ 


Therefore  ,t .  =  A./JffiL 

V  m  —  i' 

and  from  (21),  r  =  .6745 

From  (25)  and  (26),  e0  = 


m  — 


in(in  — 

=  .6745  /  ~w 


(27) 


n  —  \) 
Combining  equations  (27)  and  (24),  we  readily  find 

s  =  1.2533       *-"*"  ^L;       r 

*.  =  I-2533  — 7^         =;       ^o  =  0.8453 


=  0.8453  _ 

\m(m  — 


(28) 


In  these  expressions  [+  v\  represents  the  sum  of  the  residuals 
all  taken  with  the  positive  sign. 

These  simple  formulae  (27)  and  (28)  are  of  great  practical 
value.  When  the  number  of  observations  is  not  large  the 
values  given  by  (27)  will  be  a  little  more  accurate  than  those 

*  From  what  precedes  we  see  that  this  assumption  would  be  rigorously  true  if 
the  number  of  observations  were  infinite. 


22  LEAST  SQUARES.  §  l6. 

by  (28),  but  when  the  number  is  large  (28)  will  be  sufficiently 
accurate  for  practical  purposes,  and  the  facility  with  which 
they  are  applied  is  something-  in  their  favor. 


Probable  Error  of  the  Sum  or  Difference  of  Two  or  More 
Observed  Quantities. 

16.  Let  us  next  suppose  the  unknown  quantity  x,  instead 
of  being  directly  observed,  to  be  the  sum  or  difference  of  two 
or  more  quantities  whose  values  are  obtained  by  direct 
measurement  ;  viz.  : 

Let  x  =  y^  ±  y»  in  which  yl  and  y^  are  independent  of  each 
other  and  whose  values  are  directly  observed. 

Let  the  individual  errors  of  observation  be  — 

For/,,     J/f  J/',  .  .  .  A  7; 


The  errors  of  the  individual  determinations  of  x  will  then  be 

(A-  ±  A'},  (A-  ±  Jf"),  .  .  .  (A  -  ±  4~); 

and  if  £  is  the  mean  error  of  a  determination  of  x,  we  shall 
have 

m?  =  (A!  ±  A^J  +  (4"  ±  4,")'  +  .  .  .  +  (4W  ±  ^D2. 
Expanding  and  making  use  of  the  symbol  for  summation, 

*•«•   *    [^^J    ±    2[4,  JJ  +  [//,<]. 

Let  £,  and  f2  =  the  mean  errors  of  a  measurement  of  yl  and 
^  respectively.     Then  since,  for  reasons  before  explained, 


I/-  PRINCIPLE    OF  WEIGHTS.  2$ 


the  middle  term  ([z^z/J)  may  be  regarded  as  vanishing  in 
comparison  with  [AA]  an<3  [4A]>  we  shall  have 


or  e  =     e>+&      ......     (29) 

In  a  manner  precisely  similar  we  may  extend  the  method 
to  the  sum  or  difference  of  any  number  of  observed  quanti- 
ties, so  that  in  general  if  we  have  x  =  yl  ±  y^  ±  .  .  .  ±  JTO> 
the  mean  errors  being  respectively  e,  elt  £v  .  .  .  £m,  we  shall 
have 


e  =  Vt*  +  *;  +  £.'  +  .  .  .  +  C  =  f  .     .     (30) 

Suppose  next  that  we  have  x  —  alyl  ±  <*2jj/2  ±  .  .  .  ±  fxmymi 
in  which  or,,  or,,  .  .  .  <^m  are  constants.  If,  as  before,  e,,  ea,  .  .  . 
em  are  the  mean  errors  of  y^  yv  .  .  .  ym,  then  the  mean  errors 
of  a^,  a^yv  .  .  .  amym  will  be  respectively  a^l9  a^,  .  .  .  cxmemy 
and  the  mean  error  of  x 


-  ^[«2f3].  •  (so 


Principle  of  Weights. 


17.  In  the  foregoing  we  have  assumed  all  the  observations 
considered  to  be  equally  trustworthy,  or,  as  it  is  expressed 
technically,  of  equal  weight.  As  will  readily  be  seen,  we 
shall  frequently  have  occasion  to  combine  observations  of 
different  weights.  It  is  therefore  important  to  ascertain 
how  to  treat  them,  so  that  each  shall  have  its  proper  influ- 
ence in  determining  the  result. 

Confining  our  discussion  for  the  present  to  the  case  of  a 
directly  observed  quantity,  the  most  elementary  form  of  the 


24  LEAST  SQUARES.  §  1 7. 

problem  will  be  that  where  the  quantities  combined  are  them- 
selves the  arithmetical  means  of  several  observations  of  the 
weight  unity.  Thus,  suppose  the  quantity  x  to  be  deter- 
mined from  m'  such  observations  ;  the  most  probable  value 
of  x'  will  then  be 

,     _  */  +  */  +  <  +  .  .  .  +  nj 


From    a  second,  third,  etc.,  series  of  m",  m"' ',  etc.,  observa- 
tions we  have  respectively 


_ 


_ 


Combining  all  these  individual  values,  we  have  for  the 
most  probable  value  of  x 


-  w"  -f- 
m"x" 


The  value  of  x  will  not  be  affected  if  we  multiply  both  nu- 
merator and  denominator  of  this  fraction  by  any  constant  a  ; 
viz., 


_ 


am'  +  am"  +  am"'  +  .  .  . 


•••  /       N 


§  I/-  PRINCIPLE   OF    WEIGHTS.  2$ 

in  which  we  may  regard  am1,  am" ,  etc.,  as  the  respective 
weights  of  x1 ',  x" ,  etc.  a  may  be  integral  or  fractional. 
From  this  we  see  that  the  weights  are  simply  relative  quan- 
tities and  are  in  no  case  to  be  regarded  as  absolute. 

From  the  foregoing  we  have  the  following  practical  rule : 
When  observations  are  to  be  combined  to  which  different  weights 
are  to  be  ascribed,  the  most  probable  value  of  the  unknown  quantity 
will  be  obtained  by  multiplying  each  observation  by  its  weight, 
and  dividing  the  sum  of  the  products  by  the  sum  of  the 
weights. 

It  is  clear  that  the  difference  of  weights  may  result  from 
a  variety  of  causes  other  than  the  simple  one  considered 
above ;  as,  for  instance,  one  series  of  observations  may  be 
made  with  a  more  accurate  instrument  than  another,  or  by  a 
more  skilled  observer.  Thus,  for  example,  it  may  be  the 
case  that  ten  measurements  made  by  one  observer  will  have 
as  much  value  as  twenty  made  by  another.  If  the  weight  of 
an  observation  of  the  first  series  be  unity,  one  of  the  second 
would  only  be  entitled  to  a  weight  of  one  half ;  or  more  gen- 
erally, 

Letting/  —  the  weight  of  an  observation  of  the  second  series, 
Then  2/  =  the  weight  of  an  observation  of  the  first  series. 

If  then  we  have  a  series  x»  x»  x»  etc.,  of  observations  of 
the  weights /p/,,,/3,  etc.,  and  consequently 

x  _  A*i  +  A*.  +  A*.  +  -  •  • 


A +A+A+ - • 

as  the  most  probable  value  of  x,  it  is  evident  that,  whatever 
may  have  been  the  cause  of  this  difference  of  weight,  we  may 
consider  each  value  x^  x» etc.,  as  derived  from/!,/,,  etc.,  in- 
dividual observations  of  the  weight  unity.  Let 


26  LEAST  SQUARES.  §  1 8. 

e  =  the  mean  error  of  an  observation  of  the  weight  unity  ; 
elf  ea,  etc.,  the  mean  errors  of  xlt  x»  etc. 

Then  from  (25),     sl  =  — — ,     ?a  —  — — ,  etc., 

^.  *>._      _  k    •     (33) 

The  whole  number  of  observations  being  equal  to  pl  +  A 
H~  A  +  •  •  •  —  W  observations  of  the  weight  unity  or  of  the 
mean  error  e,  we  have  for  the  mean  error  of  x,  from  (25), 


(34) 


The  Probable  Error  when  Observations  have  Different  Weights. 

18.  The  mean  taken  according  to  weights,  as  in  equation 
(32)  or  (32)*,  is  sometimes  called  the  General  Mean.  In  order 
to  derive  the  formula  for  the  probable  error  in  this  case,  let, 
as  before,  d  be  the  error  of  the  general  mean  x0\  viz.,  x  —  XQ—  8. 
Then,  the  notation  being  as  before,  we  have 

4  =  v,  -  d,     A,  =  v,-  fi,     A,  =  vz-  d,  etc. 

The  error  A^  belongs  to  x^  and  therefore  appears/,  times ; 
The  error  Ja  belongs  to  x^  and  therefore  appears  /a  times ; 


Therefore      |>4J]  =  [pvv\  —  2^pv~\d  +  [p~]d\ 


For  the  same  reason  as  in  previous  cases  [pv]  may  be  dis- 
regarded as  being  inappreciable  in  comparison  with  the  other 
terms,  when  we  have 


§  1  8.  OBSERVATIONS  OF  DIFFERENT  WEIGHTS.        ' 

Substituting  for  d  the  mean  error  of  x  from  (34),  we  have 


Now,  as  x^  is  equivalent  to/t  observations  of  weight,  unity, 
there  will  be  the  equivalent  of  pl  errors  equal  to  A^  ;  and  el 
being  the  mean  error  of  x^  we  shall  have 


Whence  from  (33), 
Similarly, 


*?  — 


etc. 


And  m  being  the  whole  number  of  quantities,  or  observa- 
tions, xlt  xv  etc.,  we  have 


Our  equation  therefore  becomes  m£  —  [pvv]  +  c2,  from  which 


/  C/H  . 

~~\l  m-  i' 


and  from  (34), 
and  from  (21), 


£.  — 


r.  =  -6745 


/ 

Vi 


(*  - 


•    •    •    (35) 


^  in  these  formulae  is.the  number  of  individual  observations, 
or  quantities,  xlt  xv  etc.,  and  must  not  be  mistaken  for  the 
sum  of  the  weights. 

It  will  be  evident  upon  a  careful  comparison  of  these  ex- 
pressions with  the  formulae  (27)  that  we  should  have  reached 


28        •  LEAST  SQUARES.  §  19. 

thefsame  result  by  multiplying  each  quantity  x»  xv  etc.,  by 
the  square  root  of  its  weight,  and  then  proceeding  exactly  as 
we  have  previously  done  with  observations  of  equal  weight. 

We  have  therefore  established  the  following  rule  which  we 
may  apply  in  combining  observations  of  different  weights  : 

First  reduce  all  observations  to  a  common  unit  of  weight  by 
multiplying  each  by  the  square  root  of  its  weight,  then  combine 
t/iem  precisely  as  if  they  had  originally  been  of  equal  weight. 

For  examples  of  the  application  of  the  formulae  see  pages 
515  and  516. 

General  Remarks. 

19.  We  have  hitherto  considered  only  those  cases  where 
the  unknown  quantity  is  derived  in  the  simplest  manner  from 
observation,  viz.,  by  direct  measurement  or  by  the  sum  or 
difference  of  directly  measured  quantities. 

Before  proceeding  to  the  more  complex  cases  a  few  general 
remarks  may  not  be  out  of  place. 

Equation  (13),  which  represents  the  law  of  distribution  of 
error,  and  on  which  the  subsequent  discussion  is  based,  rests 
upon  two  hypotheses  neither  of  which  is  ever  fully  realized 
in  practice,  viz.,  that  the  number  of  observations  is  infinite, 
and  that  they  are  entirely  free  from  constant  errors,  i.e., 
errors  which  affect  all  alike.  The  formulse  deduced  when 
applied  to  the  cases  which  actually  arise  can  give  us  only 
approximate  results,  although  they  will  be  the  best  attainable 
approximations  from  the  given  data.  This  is  particularly  to 
be  borne  in  mind  when  the  number  of  observations  is  small. 
The  probable  errors  in  such  cases  are  apt  to  be  entirely  illu- 
sory, and  in  general  are-  only  reliable  when  the  number  of 
observations  is  large  enough  to  exhibit  approximately  the 
law  of  distribution  of  error  derived  from  the  hypothesis  of 
an  infinite  series  of  observations. 


§20.  COMPARISON  WITH   OBSERVATION.  29 

The  second  hypothesis  mentioned  above,  viz.,  that  con- 
stant errors  do  not  exist  in  our  data,  can  never  be  fully  realized, 
and  this  fact  is  often  the  source  of  great  annoyance  and  un- 
certainty in  combining  observations  taken  under  different 
conditions.  Such  errors  arise  from  a  variety  of  causes,  some 
easy  to  investigate  and  others  not  at  all  so.  It  is  of  very 
frequent  occurrence  that  a  result  derived  from  a  single  series 
of  observations  will  give  a  small  probable  error,  and  yet  differ 
widely  from  that  derived  from  a  second  series  to  all  appear- 
ances  equally  good.  It  sometimes  happens  that  computers 
who  are  puzzled  by  such  occurrences  attribute  the  difficulty 
to  faults  in  the  method,  the  truth  being  that  they  are  due  to 
the  presence  of  a  class  of  errors  with  which  the  method  does 
not  profess  to  deal. 

The  remedy  for  this  difficulty  is  to  vary  as  much  as  pos- 
sible the  conditions  under  which  the  observations  are  made, 
and  in  a  manner  calculated  to  eliminate  as  far  as  possible 
those  constant  errors  which  cannot  be  investigated. 


Comparison  of  Theory  with  Observation. 

20.  The  test  of  theory  is  its  agreement  with  observed  facts. 
We  may  in  this  manner  test  the  truth  of  the  law  which  we 
have  derived  for  the  distribution  of  errors. 

We  have  the  probability  that  an  error  falls  betwedn  the 
limits  ±  a  expressed  by  the  equation 


/+«     h 
r 


In  accordance  with  the  theory  of  probabilities,  /  here  is  a 
fraction  which  expresses  the  ratio  of  the  number  of  errors 


30  LEAST  SQUARES.  §  2O. 

between  ±  a  to  the  whole  number.     If  then  the  number  of 
observations  is  m,  the  number  of  errors  between  ±  a  will  be 


m  — — 


To  test  the  law  expressed  by  this  formula  we  have  only  to 
compute  the  probable  error  of  the  series  of  observations  under 
consideration  by  (27)  or  (28),  and  then  h  by  (19).  The  value 
of  the  integral  will  then  be  obtained  from  Table  I.,  and  we 
shall  be  in  possession  of  everything  necessary  for  comparing 
the  number  of  errors  between  any  two  limits  as  indicated  by 
this  formula  with  the  number  shown  by  the  series  of  observa- 
tions. Many  such  comparisons  have  been  made,  and  always 
with  satisfactory  results,  when  the  number  of  observations 
compared  has  been  large.  A  perfect  agreement  is  of  course 
not  to  be  looked  for,  as  our  formula  has  been  derived  on  the 
theory  of  an  infinite  number  of  observations  ;  and  further,  we 
are  not  in  possession  of  the  true  errors  for  comparison  with 
the  formula,  but  the  residuals  instead,  which  will  always  differ 
from  the  errors  unless  we  are  in  possession  of  the  absolutely 
true  value  of  the  unknown  quantity. 

As  an  illustration  of  the  above  the  following  tabular  state- 
ment gives  the  result  of  a  comparison  with  theory  of  the 
errors  of  the  observed  right  ascensions  of  Sirius  and  Altair. 
The  example  is  given  by  Bessel  in  the  Fundament  a  Astrono- 
mice. 

In  a  series  of  470  observations  by  Bradley  the  probable 
error  of  a  single  observation  was  found  to  be  r  =  o".2637, 
whence  h  —  1.80865.  Therefore  for  the  number  of  errors  less 
than  ".i  the  argument  of  Table  I.  will  be  t  —  hA  —  .180865. 
With  this  argument  we  find  for  the  integral  .20188,  which 
multiplied  by  470,  the  entire  number  of  errors,  gives  95  as 


20. 


COMPARISON  WITH  OBSERVATION. 


the  number  of  errors  less  than  ".i.     In  a  manner  similar  to 
this  the  following-  results  were  found  : 


Between 

No.  of  Errors 
by  Theory. 

No.  of  Errors 
by  Experience. 

o'  .0  and  o"i.  I 
o  .1  and  o'  .2 

95 
89 

81 

o  .2  and  o'  .3 

78 

78 

o'  .3  and  o'  .4 

64 

58 

o'  .4  and  o'  .5 

50 

51 

o'  .5  and  o'  .6 

36 

36 

o'  .6  and  o'  .7 

24 

26 

o'  .7  and  o'  .8 

15 

14 

o'  .8  and  o'  .9 

9 

10 

o'  .9  and  i'  .0 

5 

7 

over     i'  .0 

5 

8 

This  agreement  is  very  satisfactory,  but  here,  as  in  other 
similar  examples,  the  larger  errors  occur  a  little  more 
frequently  than  theory  would  indicate. 

This  is  probably  due  to  the  fact  that  (unconsciously,  per- 
haps)  every  observer  will  occasionally  let  an  observation  pass 
which  is  not  up  to  the  average  standard  of  accuracy.  Small 
mistakes  will  sometimes  occur,  also,  which  are  not  of  sufficient 
magnitude  to  attract  attention.  A  consideration  of  the  matter 
has  led  to  attempts  on  the  part  of  Peirce  of  Harvard  College 
and  Stone  of  England  to  establish  criteria  for  the  rejection 
of  such  doubtful  observations.  On  the  other  hand  it  has  been 
proposed  to  overcome  the  difficulty  by  determining  a  system 
of  weights  which  should  give  those  observations  which  show 
large  discrepancies  less  influence  than  those  showing  small 
ones. 

This  branch  of  the  subject,  however,  is  beyond  the  scope 
of  the  present  work.  It  is  an  exceedingly  delicate  matter 
to  deal  with,  and  from  its  nature  is  probably  incapable  of  a 
mathematical  treatment  which  shall  be  entirely  satisfactory. 

Every    computer   occasionally  feels  compelled   to   reject 


32  LEAST   SQUARES.  §21. 

observations.  This  should  always  be  done  with  extreme  cau- 
tion. As  for  the  criteria  for  this  purpose  hitherto  proposed, 
probably  the  most  that  can  be  said  in  their  favor  is  that  their 
use  insures  a  uniformity  in  the  matter,  thus  leaving  nothing 
to  the  individual  caprice  of  the  computer. 

Indirect  Observations. 

21.  We  have  now  investigated  the  simplest  case  of  the 
determination  of  the  unknown  quantity  by  observation,  viz., 
that  when  the  quantity  to  be  determined  is  measured  directly. 
In  the  more  general  form  of  the  problem  the  unknown 
quantities  are  connected  with  the  observed  quantities  by  an 
equation  of  the  form 


M  being  given  by  observation,  and  x,  y,  2,  etc.,  being  the  un- 
known quantities.  This  general  form  includes  the  case  which 
we  have  previously  investigated,  where  there  was  only  one 
unknown  quantity.  Each  observation  furnishes  an  equation 
of  this  form  ;  therefore  a  number  of  observations  equal  to  that 
of  the  unknown  quantities  will  completely  determine  their 
value. 

This  would  leave  nothing  to  be  desired  if  the  observations 
were  perfect  ;  but  owing  to  the  errors  to  which  they  are  liable, 
the  values  of  x,  y,  2,  etc.,  will  be  more  reliable  the  greater 
the  number  of  observations  on  which  they  depend.  If  now 
we  have  four  unknown  quantities,  x,y,  z,  and  w,  four  observa- 
tions will  give  us  four  equations  from  which  the  values  of  the 
unknown  quantities  may  be  determined.  If  we  have  more 
than  four  equations,  we  may  determine  values  of  the  unknown 
quantities  by  combining  any  four  of  them.  As  the  equations 
depend  on  observations  more  or  less  erroneous,  we  should 
thus  obtain  a  variety  of  values  for  x,  y,  z,  and  w,  all  of  them 
probably  in  error  to  some  extent. 


§21.  IN  DIRE  CT   OB  SER  VA  7  'IONS.  3  3 

The  problem  then  is  this  :  Of  all  possible  systems  of  values 
of  the  unknown  quantities,  to  find  that  which  most  accurately 
represents  all  of  the  observations. 

We  shall  confine  ourselves  to  the  consideration  of  linear 
equations  ;  and  as  the  problems  in  which  we  shall  be  more 
particularly  interested  do  not  give  rise  to  equations  of  more 
than  four  unknown  quantities,  we  shall  limit  our  discussion  to 
that  number.  It  will  be  obvious,  however,  that  it  can  be 
extended  to  any  number. 

Suppose  we  have  the  following  system  of  equations  : 


,y  +  c,z  +  d,w  = 
,  y  +  c,z  +  djv  =  »,; 


in  which  x,  y,  z,  and  w  are  unknown  quantities,  a,  b,  c,  d, 
etc.,  are  coefficients  given  by  theory,  and  «„  nv  n^  etc.,  are 
quantities  given  by  observation. 

If  now  the  data  were  perfect  we  should  obtain  the  same 
values  of  x,  y,  z,  and  w  by  combining  any  four  of  these 
equations.  Owing,  however,  to  the  errors  of  observation  to 
which  nlt  «„,  etc.,  are  subject,  it  is  not  probable  that  a  substitu- 
tion of  the  true  values  of  x,  y,  z,  and  w  (if  we  knew  them) 
would  exactly  satisfy  any  one  of  the  equations. 

Let  v^  v»  v»  etc.,  be  the  residuals  obtained  by  substituting 
in  equations  (36)  for  x,  j,  z,  and  w  their  approximate  values 
such  that  the  following  equations  will  be  rigorously  satisfied  : 


=  n,  — 
-f-  c^z  -\-  djw  ?=  HI  —  v^.\  •     •     •     (37) 


34 


LEAST  SQUARES. 


21. 


Now  the  most  probable  values  of  our  unknown  quantities 
will,  be  those  which  make  the  sum  of  the  squares  of  these 
residuals  a  minimum,  viz., 


etc.  = 


•     (38) 


must  be  a  minimum. 

In  these  equations  x,  y,  z,  and  w  are  supposed  independent, 
therefore  the  differential  coefficients  with  reference  to  each 
variable  must  separately  be  equal  to  zero  to  satisfy  the 
conditions  of  a  minimum.  That  is, 


d\yv\  _ 


=  o, 


d\vv~\  _ 


=  o, 


d\vv\ 


—  o. 


dx  dy  dz  dw 

Writing  out  these  expressions  in  full,  we  have  the  following: 

dvl  dv^    .        dvz    . 


dv. 


— 3+        =o- 

dv« 


Y-   •    •    (39) 


,y,  2,  and  w  being  independent,  we  have  from  (37), 


£--* 


dv* 

' 


dj- 

*:W-4* 


|=->.,  etc. 


—  — 
dw 


-r— 
dw 


%  _  d  .  . 
-=  --  -  «3,  etc., 
dw 


§21.  INDIRECT  OBSERVATIONS. 

by  means  of  which  values  equations  (39)  become 

'  av     -\-    aV     -\-    av     Ji-    %        .    =    O- 
•"'I"!        |          "22        I        *"   33        I  *  **  I 


35 


jl    ::^....(4o) 


.  .  .  =  o . 


Substituting  for  ^,  va,  etc.,  their  values  from  (37),  we  have 
for  the  first  of  these 

a,a,x  +  a  fay  +  a.c^z  +  a&w  -  a,n,  -^ 


+  a,b,  y  +  a,c,z  + 

+  a,b,  y  +  ajj  +  a&w  —  asn, 


-  =  O. 


The  second  of  (40)  becomes 


and  similarly  for  the  remaining  equations.    .Using  Gauss' 
symbols  of  summation,  we  have  therefore 

\ad\x  -f-  \_a&\y  +  \_ac\z  -f-  \ad~\w  =  \ari\ ;  -^ 
[«*>  +  [W]  j  +  [fc>  +  [W]w  =  \bn\ ;  I 
[«>  +  \bc\y  +  \cc-\z  +  \cd-\w  =  [«,];  f" 


These  are  called  Normal  Equations,  and  the  values  of  the 
unknown  quantities  obtained  by  solving  them  will  be  the 
system  of  values  which  makes  the  sum  of  the  squares  of  the 
residuals  vlt  v»  etc.,  a  minimum,  and  therefore  the  most  prob- 
able system  of  values.  Equations  (36)  are  called  Equations  oj 


36  LEAST   SQUARES.  §  22. 

Condition,  or  Observation  Equations.  An  inspection  of  (41) 
gives  us  the  following  rule  for  solving  a  series  of  equations 
of  condition  : 

Multiply  each  equation  by  the  coefficient  of  x  in  that  equation, 
then  add  together  the  resulting  equations  for,  a  new  equation, 
then  multiply  each  equation  by  the  coefficient  of  y  in  that  equation, 
and,  as  before,  form  the  sum  of  the  resulting  equations.  Continue 
the  process  with  the  coefficients  of  each  of  the  unknown  quantities. 
The  number  of  resulting  Normal  Equations  will  be  equal  to  tJiat 
of  the  unknown  quantities,  and  the  values  of  the  unknown  quanti- 
ties deduced  therefrom  will  be  the  most  probable  values. 

It  must  be  borne  in  mind  that  this  process  supposes  the 
number  of  equations  of  condition  to  be  greater  than  that  of 
the  unknown  quantities.  If  it  is  less,  this  process  will  give 
us  a  number  of  equations  equal  to  that  of  the  quantities  to  be 
determined,  but  they  will  be  indeterminate  none  the  less  than 
the  original  equations  were,  as  can  be  easily  shown. 

v    Observations  of  Unequal  Weight. 

22.  In  deriving  the  normal  equations  from  the  equations 
of  condition,  we  have  regarded  the  latter  as  of  equal  weight. 
In  the  more  general  case  the  weights  will  be  unequal. 

In  the  equation  a,x  +  b^y  +  c,z  +  djv  —  n,,  if  we  suppose, 
as  in  (33),  that  p,  represents  the  weight  of  an  observation, 
viz.,  of  »„  that  e,  is  the  mean  error  of  «„  and  e  the  mean  error 
of  an  observation  of  weight  unity,  we  have 


Multiplying  the  above  equation  by  Vp»  we  have 

a,  Vps  +  b,  Vp.y  +  c,  Vp,z  +  d,  Vp,w  =  «,  Vp,,      (42) 


§23.  ARRANGEMENT  OF  COMPUTATION.  37 

an  equation  in  which  the  mean  error  of  the  absolute  term 
ni  ^A  ls  f>  anc*  tne  weight  unity.  In  the  same  manner  we 
multiply  each  equation  by  the  square  root  of  its  weight,  thus 
reducing  them  all  to  the  same  unit  of  weight,  when  we  pro- 
ceed precisely  as  before  in  forming  the  normal  equations. 


Computation  of  the  Coefficients. 

23.  The  method  of  forming  the  normal  equations  is  now 
fully  explained;  the  work  of  computation,  however,  is  some- 
what laborious,  especially  when  the  number  of  equations  of 
condition  is  large.  It  will  therefore  be  important  to  arrange 
the  work  so  that  the  numerous  multiplications  and  additions 
may  be  performed  with  the  least  liability  to  error,  and  so 
that  convenient  checks  may  be  applied  for  insuring  accuracy 
in  the  results.  The  multiplications  may  be  performed  by 
logarithms,  in  which  case  a  four-place  table  will  give  the 
necessary  degree  of  precision,  or  Crelle's  multiplication-table 
may  be  employed  with  advantage.*  We  shall  also  show 
how  to  perform  the  multiplications  by  the  use  of  a  table  of 
squares. 

Convenient  proof-formulas  may  be  derived  as  follows :  Let 
the  sum  of  all  the  coefficients  entering  into  each  equation  be 
formed  in  succession,  and  represent  them  by  s  with  the  proper 
subscript.  Thus : 


*  Dr.  A.  L.  Crelle's  "  Rechentafeln  welche  alles  multipliciren  und  dividiren  mit 
Zahlen  unter  Tausend"  (Berlin,  1869). 


38  LEAST  SQUARES.  §  24. 

Multiplying  these  sums  by  their  respective  a,  b,  c,  etc.,  in 
succession,  and  adding  the  products,  we  shall  have  the  follow- 
ing equations  for  checking  the  accuracy  of  the  coefficients  of 
the  normal  equations : 


[aa]  +  [off]  +  \ac\  +  \ad]  -  [an]  =  [as]  ; 
[off]  +  \b6\  +  \bc\  +  \bd-\  -  [M]  =  [fo] ;  . 
[ac\  +  \bc\  +  \cc\  +  \cd}  -  \cn\  =  [cs] ;  r 
[«(]  +  [&/]  +  [«f ]  +  [<&/]  -  [</»]  = 


This  requires  the  computation  of  the  additional  terms  [as], 
[t>s],  .  .  .  and  the  agreement  must  come  within  the  limit  of 
error  of  the  computation.  These  additional  terms  will  be 
further  useful  for  checking  the  accuracy  of  the  solution  of 
the  normal  equations,  as  will  afterwards  appear. 

24.  If  it  should  happen  that  the  coefficients  of  one  unknown 
quantity  in  the  equations  of  condition  were  much  larger  than 
those  of  another,  considerable  discrepancies  might  exist  in 
the  agreement  of  the  proof-formulas  with  the  sums  of  the  co- 
efficients. It  will  generally  be  necessary  practically  to  limit 
the  computation  to  a  certain  number  of  decimals,  when  the 
products  of  the  large  quantities  may  introduce  errors  into 
the  last  places,  where  the  products  of  the  small  quantities 
introduce  none. 

This  difficulty  is  overcome  by  substituting  for  the  unknown 
quantities  other  quantities  which  will  make  the  coefficients 
of  the  same  order  of  magnitude  throughout.  This  is  con- 
veniently accomplished  by  selecting  the  largest  coefficient 
with  which  an  unknown  quantity  is  affected  and  dividing 
.each  of  the  coefficients  of  this  quantity  by  it.  Thus,  let 
a,  ft,  y,  d  be  the  largest  coefficients  of  the  quantities  x,y,  z,  w, 
respectively,  which  occur  in  the  equations  of  condition,  and 
let  v  be  the  largest  of  the  series  of  known  quantities  nlt  n» 


§25.  ARRANGEMENT  OF  COMPUTATION.  39 

#3,  .  .  .     Then  we  may  place  the  equations  of  condition  in  the 
following  form  : 


where  the  unknown  quantities  are  (<*.*•),  (/#?),  .  .  .  and  the 
values  obtained  in  solving  the  equations  will  be  in  terms  of 
v.  The  equations  will  be  made  homogeneous  by  this  pro- 
cess before  beginning  the  work  of  forming  the  normal  equa- 
tions. The  sums  slt  Ja,  .  .  .  will  be  mo^t  convenient  for  the 
purpose  to  which  they  are  applied,  if  they  are  formed  from 
these  homogeneous  equations. 

For  the  Jdnd  of  problems  which  we  shall  have  occasion  to 
solve  in  the  following  pages  there  will  seldom  be  a  system- 
atic difference  in  the  magnitudes  of  the  coefficients  of  the 
different  unknown  quantities  of  importance  enough  to  render 
this  operation  necessary.  In  cases,  however,  where  there  is 
a  marked  difference  in  this  respect  it  will  be  advisable  to 
incur  the  slight  additional  labor  involved,  and  in  some  cases 
it  becomes  a  matter  of  considerable  importance. 

25.  The  formation  of  the  normal  equations  with  the  accom- 
panying proof-formulae  will  therefore  require  the  computa- 
tion of  the  following  quantities  : 

[aa]  [a6]  [ac]  [act]  [an]  [as]; 
\bb\  [be}  [bd]  [6n]  [6s]; 


[nn]  [ns]  . 


4°  LEAST  SQUARES.  §  25. 

The  latter  will  be  employed  for  checking  the  final  compu- 
tation, as  will  be  shown  hereafter.  As  will  be  seen,  there  are 
twenty  of  these  quantities  required  in  a  series  of  four  equa- 
tions. In  general  the  number  will  be*  <>  +  2)  («  +  3)  _  ^ 

where  n  is  the  number  of  unknown  quantities. 

Let  a  sheet  of  paper  be  ruled  with  a  number  of  vertical 
columns  represented  by  the  above  formula.  In  the  first 
horizontal  line  will  be  the  symbols  of  the  products  written  in 
the  columns  below,  viz.,  [aa],  [ad],  .  .  .  and  in  the  last  line  the 
sums  of  the  products.  If  the  results  are  correct  the  proof- 
equations  (44)  must  be  satisfied.  The  algebraic  signs  of  the 
various  products  will  demand  special  attention,  as  they  form 
a  very  fruitful  source  of  error. 

If  the  application  of  the  proof-formulae  is  postponed  until 
the  conclusion  of  this  part  of  the  computation,  the  position 
of  an  error  is  often  shown  at  once,  since  each  sum,  with  the 
exception  of  the  sum  of  the  squares,  is  found  in  two  different 
proof-equations.  If  two  of  the  proof-formulas  fail  to  be 
satisfied,  while  the  others  prove  true,  the  error  is  in  the  term 
common  to  both  ;  while  if  only  one  equation  fails  to  be  satis- 
fied, the  error  is  in  the  quadratic  term. 

Before  proceeding  further  it  is  recommended  .that  the 
reader  refer  to  the  example  found  on  page  329.  The  num- 
ber of  observation  equations  is  twelve,  each  of  which  has 
been  multiplied  by  the  square  root  of  its  weight.  The  num- 
ber of  unknown  quantities  is  three,  the  coefficients  of  which 
have  no  systematic  difference  in  magnitude  of  sufficient 
importance  to  require  the  application  of  the  process  for 
rendering  them  homogeneous.  The  formation  of  the 
normal  equations  is  found  on  page  330.  The.  number  of 


*  It  is  the  sum  of  a  series  of  terms  in  arithmjtical  progression  minus  i;  num- 
ber of  terms  =  (n  -\-  2);  first  term  =  i;  last  term  =  (n  -f-  2). 


§  26.  ARRANGEMENT  OF  COMPLICATION.  4! 

unknown  quantities  being  three,  we  require  by  the  formula 
just  given  fourteen  columns.  It  will  be  observed  that  the 
proof-formulas  are  perfectly  verified,  as  they  should  be  in 
this  case,  no  decimal  terms  having  been  neglected. 


Computation  of  the  Coefficients  by  a  Table  of  Squares. 

26.  By  whatever  method  the  multiplications  are  performed 
a  table  of  squares  will  be  found  very  convenient  for  the 
quadratic  terms.  Terms  of  the  form  [#A]  may  also  be  com- 
puted with  such  a  table,  as  will  appear  from  the  following. 

We  have       afr  =  £{(*,  +  b^  -  a?  -  b?}\ 


•     •     (45) 

The  quadratic  terms  \ad\,  \bb\,  .  .  .  will  be  computed  in  any 
case,  so  there  will  only  be  required  in  addition  the  terms  of 
the  form  [(a  +  £)a].  In  case  of  four  unknown  quantities  we 
shall  require  the  following  quadratic  terms : 


\ad\  [(*  +  £)3]  [(*  +  ^)2]  [(a  +  </)']  [(a  -  nf]  ;  ] 
\bU\        \_(b  +  ;)2]  [_(b 


[ss]  [nn\. 


(46) 


The  last  two  will  be  employed  in  checking  this  and  the  sub- 
sequent computation.  Thus  for  the  case  of  four  unknown 
quantities  we  have  sixteen  terms  of  the  above  form,  or  in 

(n  _J_  j)  („  _j_  2)    , 
general,  i-       -J-±-  l — I  -f-  i. 


LEAST  SQUARES. 


26. 


The  equations  having  been  multiplied  by  the  square  roots 
of  their  respective  weights,  and  the  coefficients  made  homo- 
geneous if  necessary,  the  computation  will  be  carried  out  as 
shown  in  the  following  scheme : 


bb 


6,6, 


(a  +  bj* 


[««]  f  [ 

•ra 
W 


(a  +  c)* 


(b  + 


t(*  4- 


an]      '          a\6c] 
an\  [Sc] 


In  order  to  derive  a  convenient  proof-formula  we  square 
both  members  of  equations  (43)  and  add 


[ss]  +  3  iM  + 


+  \cc\  + 
']  +  [(«  f 

]  +  [(^  + 
+  \.(c  + 


+  [»»]}  = 


-  »)']- 


(47) 


For  an  example  bf  the  application  of  the  above  method 
the  reader  will  turn  to  page  334,  where  the  normal  equations 
are  computed  from  the  equations  of  condition  before  re- 
ferred to.  This  method  possesses  some  advantages  over 
that  by  direct  multiplication:  the  most  important  of  these  is 
in  the  fact  that  the  liability  to  error  in  algebraic  signs  is  for 
the  most  part  avoided.  Care  being  taken  in  forming  the  sums 
(a  _)_  £),  (a  -j-  c\  etc.,  no  further  attention  need  be  given  to 
the  algebraic  signs  until  the  coefficients  of  the  normal  equa- 
tions are  completed. 


§  27.  SOLUTION  OF  NORMAL  EQUATIONS.  43 


Solution  of  the  Normal  Equations. 

27.  In  the  solution  of  the  normal  equations  the  work  should 
be  arranged  so  that  it  may  be  conveniently  reviewed   for 
detecting  errors  in  case  such  exist,  and  so  that  proof-formulas 
may  be  applied  at  the  various  stages  of  progress. 

The  order  in  which  the  unknown  quantities  are  determined 
is  generally  indifferent  except  in  the  case  where  the  nature 
of  the  problem  is  such  that  one  or  more  of  them  cannot  be 
determined  with  accuracy  from  the  equations.  We  may 
know  in  advance  that  we  have  a  case  of  this  kind,  or  it  may 
be  discovered  in  solving  the  equations. 

It  will  be  shown  hereafter  that  the  weight  of  any  unknown 
quantity  will  be  determined  by  arranging  the  solution  in  such 
a  way  that  this  quantity  is  determined  first.  The  weight  will 
then  be  represented  by  its  coefficient  in  the  last  equation  from 
which  the  others  haVe  been  eliminated.  If  now  this  coefficient 
is  very  small  it  shows  that  this  quantity  cannot  be  well 
determined  without  additional  data,  and  the  solution  must 
then  be  arranged  so  that  the  uncertainty  in  this  quantity  will 
have  the  least  effect  on  the  others.  In  case  a  preliminary 
computation  shows  that  the  weight  of  any  unknown  quantity 
is  very  small,  the  elimination  will  be  repeated  in  such  a  way 
that  this  quantity  is  first  determined.  The  values  of  the 
others  will  then  be  expressed  in  terms  of  this  one.  If  then 
at  any  time  additional  data  become  available  for  determining 
this  quantity,  or  if  it  is  known  from  any  other  source,  the 
other  quantities  become  known  also. 

As  such  cases  will  seldom  occur  in  the  problems  with 
which  we  shall  have  to  deal,  it  will  not  be  necessary  to  enter 
more  fully  into  the  matter  at  present. 

28.  In  the  elimination  it  will  be  convenient  to  employ  the 
method  of  substitution,  using  a  form  of  notation  proposed  by 


44  LEAST  SQUARES.  §  28. 

Gauss.  In  developing  the  formulae,  we  shall  suppose  as  before 
the  number  of  unknown  quantities  to  be  four.  It  will  be  a 
simple  matter  to  extend  or  abridge  them  in  case  of  a  greater 
or  less  number. 

The  equations  to  be  solved  are 

\ad\x  -\-  \ab~\y  +  \_ac\z  +  \_ad~\w  =  [an]  ; 
\aS\x  +  \bb\y  +  \bc\z  +  \bd-\w  ==  \bn\  . 

\_ac\x  +  \bc\y  +   \cc\*  +  [cd^w  ==   \cn\  ; 
\_ad]x+  \bd~\y  +  \cd]z^\dd]w  =  \dn~\ 

From  the  first  of  these  we  have 


_ 

\aa\       \aa~Y        [aa]         \_aa] 

which  value  being  substituted  in  the  remaining  three  equa- 
tions, we  shall  have  x  eliminated.  The  first  of  the  resulting 
equations  will  be 


and  similarly  for  the  remaining  two. 
Let  us  now  write 


-          [Vi*]  =  \bb  i]  ;         \M\  -  f9[«rf]  =  \bd  I]  ;  ] 


§  28.  SOLUTION  OF  NORMAL  EQUATIONS.  45 

and  for  the  coefficients  of  the  second  equation, 


Similarly  for  the  third, 

-          ca*]  =•[*•  i]. 


Our  three  equations  then  become 

\bb  i]y  +  \bc  i\z  +  \bd\~\w  =  [bn  i]  ;  ) 

[fo  i]^  +  \cc  i\z  +  \cd  i\w  =  [«i  i]  ;  I  .  .     .     (SO) 

\bd\\y  +  [I>  +  [^i]w  =  \dn  i]  .  ) 


In  these  the  same  symmetry  of  notation  is  preserved  as  in 
the  normal  equations,  and  it  can  easily  be  shown  that  the 
terms  [bb  i],  \cc  i],  and  \_dd  i],  which  have  the  quadratid  form, 
will  always  be  positive. 

From  the  first  of  (50)  we  have 

_ 

~ 


\bb  i]       \bb  i  \bb  i 

This  is  to  be  substituted  in  the  second  and  third,  and  the  fol- 
lowing auxiliary  coefficients  computed  : 


[te I] =^cn 2] ; 


-(49), 


46  LEAST  SQUARES.  §  28. 

which  process  gives  us  the  following  equations  : 

\cc  2\z  +  \cd  2]w  =  [en  2]  ;  )  ,    . 

\cd2\z  +  \_dd2\w  =  \_dn2\.\  ' 

From  the  first  of  these, 


<»> 


Substituting  this  in  the  second,  and  writing 


•  rcc  ^Lfr 

we  have 

^J  —  L1"*  31*     L""*  ~J         \CC1Y 

\dd  ^\w  —  \dn  3]  ;    . 

(54) 

from  which 

0;0 

z,  y,  and  x  can  now  readily  be  found  by  substituting  succes- 
sively in  (53),  (51),  and  (48). 

The  first  equation  in  each  of  (41),  (50),  (52),  and  (54)  are 
called  elimination  equations,  and  are  here  brought  together 
for  convenience  of  reference  : 

\_ad\x  +  \ab~\y    +  \_ac~\z    +  \ad~\w    —  [an]  ; 

\bb  i\y  +  \bc  i>  +  \bd  i]w  =  \bn  i]  (  6 

\CC2~\Z  +  \cd2\W  =   \CH2]\ 


This  is  all  that  will  be  strictly  necessary  in  case  the  weights 
and  probable  errors  of  the  unknown  quantities  are  not  re- 
quired. 


§  29.  PROOF-FORMULAE.  47 


Proof-Formula. 

29.  Convenient  proof-formulas  for  checking  the  accuracy 
of  the  successive  auxiliary  coefficients  may  be  derived  from 
the  summation  terms  [as],  [fa],  ...  of  equations  (44). 

Referring  to  these  formulae,  let  us  write 


Substituting  for  [fa]  and  [as]  their  values,  this  expression 
may  be  written  in  the  form 


(*"]  =  [M  -  [SKI  + 


Therefore,  writing  for  the  quantities  in  the  brackets  their 
values,  we  have 

[bs  i]  =  [bb  i]  +  [be  i]  +  [&/  1]  -  [bn  i], 

a  formula  by  which  the  accuracy  of  the  coefficients  in  the 
second  member  can  be  tested,  and  which  requires  the  addi- 
tional auxiliary  quantity  [fa  i]. 

Proceeding  in  a  similar  manner,  we  shall  require  for  check- 
ing the  computation  at  the  end  of  the  first  stage  of  the  eli- 
mination the  following  auxiliary  quantities  : 

[6s  i]  =  [6s]  -  E[«]  ;        [«  i]  =  [«]  -         [«]  ; 


48  LEAST  SQUARES.  §  30. 

when  we  shall  have  the  following  proof-equations: 

[bs  i]  =  [bb  i]  +  [be  i]  +  [bd  i]  -  [bn  i]  ;  ) 
\cs  i]  -  [be  i]  +  [«  i]  +  [cd  i]  -  [en  i]  ;  [     .     (57) 
+  [cdi]  +  [dfc/i]  -  [dn  i]  .  ) 


In  the  same  manner  we  have,  for  checking  the  next  step  in 
the  operation, 


[«  2]  =  [«  I]  -  [fa  I]  ;       [A  2]  =  \ds  I]  -  [fe  I]: 


(      } 
[ds  2]  = 


(59) 


and  finally,          [^3]  =  [<&  2]  -          -J[«2]; 


The  agreement  of  these  two  values  of  [ds  3]  must  be  within 
the  limits  of  error  of  the  computation,  and  it  furnishes  a  very 
accurate  control  over  the  accuracy  of  the  computation  up  to 
this  point. 

30.  After  the  values  of  x,  y,  z,  w  have  been  determined,  a 
most  thorough  proof  of  the  accuracy  of  the  entire  computa- 
tion is  obtained  by  means  of  the  residuals,  v19  v»  .  .  .  obtained 
by  substituting  these  values  of  xty,  2,  w  in  the  equations  of 
condition,  (37),  p.  33,  viz.  : 


§  3O.  PROOF-FORMULA.  49 

Multiplying  these  equations  by  —  v^  —  v9,  —  z>3,  .  .  .  in  order, 
adding,  and  writing,  in  accordance  with  the  notation  em- 
ployed, 


we  have 

[mi]  —  [av\x  —  \bv~\y  —  \cv~\z  —  \_dv~\w  =  \yv\  ; 

V 

but  by  equations  ^40), 

[av\  —  o,        \bv\  —  o,        \cv\  =  o,        [dv\  —  o. 
Therefore  [nv\  =  \vv\  ........     (60) 

Now  multiply  equations  (37)  by  nl9  n»  ns  ...  in  order,  and 
add,  viz.  : 

\nn\  —  \ari\x  —  \_bri\y  —  \cri\z  —  \dri\w  =.\nv\  —  \yv\.  (61) 

By  means  of  this  equation  \vv\  may  also  be  computed  as 
soon  as  x,  y,  z,  w  become  known.     But  we  have 

[an]        \ab~]  \ac\          [ad] 

x  =  t  _  i  —  t  _  i  v  —  --  _  -z  —  -  _  -iii)  (A%} 

[aa]        [aaY        [aa]          [aa] 
Let  this  value  be  substituted  in  (61),  and  write 

M  "'"ra^  =  ^I]; 
also  write  [bn  i],  [en  i],  etc.,  for  their  values,  when  we  have 

[nn  i]  —  [bn  i]  y  —  [en  \\z  —  [dn  i}w  =  [vv\. 
-    Let  the  same  process  be  carried  on  for  eliminating/,  z,  and 


50  LEAST  SQUARES.  §  31. 

w  in  succession  from  this  and  the  resulting  equations.  We 
shall  have  in  all  the  following  auxiliary  quantities  to  com- 
pute: 

\nn  i]  -  [*.]  -  [g||>«]  ;  \nn  2]  =  \nn  i]  -  M[fc»  i]  ; 
[««3]  =  \nn2\  -•[«,2]i  [«.4]  =  [««3]  -  [^  3]- 


Either  of  the  following  equations  will  then  give  the  value  of 
[vv]: 

[nn]  —  \ari\x  —  \bri\y     —  \_cri\z     —  \dri\w     =  [vv]', 
[nn  i]  —  [bn  \\y  —  [en  i]z  —  [dn  i]w  —  [vv]  ; 

[nn  2]     —  [en  2]z  —  [dn  2\w  —  [vv]  ;  ^  (62) 
[nn  3]    —  [a*  3]w  =  [yv~]  ; 
[nn  4]     =  [vv]  . 

Only  the  last  of  these  will  generally  be  used. 

31.  The  value  of  [004]  =  [vv]  can  be  derived  from  the 
summation  quantities  [ns]t  [ns  i],  etc.,  with  very  little  addi- 
tional labor.  We  have 

[ns]  =  [an]  +  \bn~\  +  [en]  +  [dn]  -  [nn]. 

f  'A.  r  -i  r       n  [an]  r       -, 

Let  us  write  [ns  i]  =  [ns]  —  j: — i[as^ 

and  substitute  in  this  expression  for  [ns]  and  [as]  their  values, 
when  it  may  be  placed  in  the  following  form : 


§32.  ARRANGEMENT  OF  COMPUTATION.  JI 

or  what  is  the  same  thing, 

[ns  i]  =  [bn  i]  -f-  [en  i]  +  \dn  l]  —  \nn  i]. 

Proceeding  in  a  similar  manner  to  form  in  succession  the 
following  auxiliary  quantities,  we  have  the  series  of  equations 
by  which  the  accuracy  of  the  quantities  [bn  i],  [en  i],  .  .  . 
[nn  4]  may  be  verified  : 


Ins  ?]  -  [«  2]  -  [«  2];  [«4]  =  [«  3]-[*  3]- 


i]  =  [bn  i]  +  [V*  i]  +  \dn  i]  —  \nn  i]  ; 
2]  =  \cn  2]  +  [^  2]  —  [iw  2]  ; 


-(49)' 


(63) 


Only  the  last  of  these  equations  will  generally  be  required. 

Form  of  Computation. 

32.  In  computing  the  various  auxiliary  quantities  which 
occur  in  the  solution  of  a  series  of  normal  equations,  the  work 
should  be  arranged  so  that  it  may  be  carried  through  from 
beginning  to  end  in  a  systematic  manner  in  order  to  keep  a 
general  oversight  of  the  results  at  the  various  stages  of  prog- 
ress, and  to  apply  conveniently  the  proof-formulas.  This  will 
be  the  more  important  the  greater  the  number  of  unknown 
quantities.  The  following  scheme  will  be  found  to  answer 
these  requirements. 

It  will  generally  be  found  expedient  to  make  the  computa- 
tion by  the  use  of  logarithms,  but  in  some  cases  the  computer 
may  prefer  to  perform  the  multiplications  and  divisions  by 
the  aid  of  Crelle's  table.  In  the  following  scheme  we  have 


52  LEAST  SQUARES.  §  32. 

supposed  logarithms  used.  A  sheet  of  paper  is  first  ruled 
with  vertical  columns,  the  number  of  which  is  greater  by  two 
than  that  of  the  unknown  quantities.  In  the  first  horizontal 
line  will  be  written  in  order  the  coefficients  which  are  com- 
bined  with  a,  viz.,  [aa],  [ab],  .  .  .  [an],  [as],  and  immediately 
below  these  their  logarithms.  Attention  is  directed  to  this 
line  by  means  of  the  letter  E  in  the  margin,  as  it  is  the  first 
of  the  elimination  equations  (56),  and  will  be  used  for  deter- 
mining x  after  y,  z,  and  w  become  known. 

In  the  third  line  are  the  coefficients  [bb~],  [be],  .  .  .  [bs],  so 
placed  that  the  letters  combined  with  b  fall  in  the  same  verti- 
cal column  with  the  same  letters  combined  with  a,  viz.,  [be] 
under  [ae],  [bd]  under  [ad],  etc. 

In   the  fourth   line  of  the  first  column   is  now  written 

log  ~4,  the  value  of  which,  as  well  as  those  of  all  the  quan- 

\aa\ 

tities  in  this  column,  must  be  carefully  verified,  as  an  error 
in  this  factor  may  not  be  detected  by  the  proof-formula. 

The  log  t_J.  is  now  written  on  the  lower  edge  of  a  card 

and  added  in  succession  to  the  logarithms  of  [ab~],  [ac\,  .  .  . 
[as],  and  as  each  addition  is  performed  the  natural  number  is 
taken  from  the  logarithmic  table  and  written  in  the  place  in- 
dicated in  the  scheme.  With  a  little  practice  the  computer 
will  be  able  to  make  this  addition  mentally,  and  take  from 
the  table  the  corresponding  number  without  writing  down 
this  logarithm.  Thus  we  shall  have 

Wr  n       •*«.  ,1 

P — i[a£n  written  under 

[aa]L 

f — #ac\  written  under 


§32. 


ARRANGEMENT  OF  COMPUTATION. 


53 


log  [aa] 


lOg  7  -  ^* 
8  [aa] 


log 

* 


[«*] 

log  [a&]  log  [ac] 


log  [W  i] 


[«»  l] 


«»2 

\cn  2]  f 

?  -  i  \fn  2] 

[«   2j 


log  [<*<?  i] 


[era] 


log 


[6di\ 

log  ^^  i] 


fofft] 

log  [^  2] 


Utf] 


log  [aw] 


log  \bn  i] 


log  [f  »  2] 


log  [ 


log  [^  3] 


log 


log  [as] 


log  [^  i] 


lOg  [<TJ  2] 


II 


III' 


IV 


VII 


•CTTTT 

VIH 


Proof-Equations. 


P. 

II. 

IIP. 


IV. 
V. 

VI'. 

VII. 

IX     VIII. 
IX. 


'6s  i 


ns3 


£tt 


dd  2]  — 


dn  2 


Practically  only  those  proof-equations  which  are  distinguished  by  an  accent  will  ordinarily 
be  employed.  The  lines  marked  by  an  E  in  the  margin  give  the  logarithms  of  the  coefficients 
of  the  elimination  equations.  The  logarithms  marked  *  must  be  carefully  verified,  since  an 
trror  in  one  of  these  may  escape  detection  by  the  proof-equation. 

For  the  application  to  a  numerical  example  see  page  331. 


54  LEAST  SQUARES.  §  33. 

and  by  subtraction, 


These  are  the  coefficients  of  the  second  elimination  equation, 
and  will  be  used  for  determining  y  after  z  and  w  have  become 
known.  The  I  in  the  margin  refers  to  the  proof-formula 
by  which  the  values  of  these  quantities  will  be  verified. 

It  will  not  be  necessary  to  proceed  farther  with  this  ex- 
planation, as  a  reference  to  the  scheme  in  connection  with 
the  formulae  for  the  auxiliary  quantities  will  show  clearly  the 
process.  The  elimination  being  completed,  the  quantities 
[**4]  and  —  \ns^\  are  computed  as  shown  in  the  scheme,  the 
agreement  of  which  with  each  other  and  with  \yv\,  obtained 
by  substituting  the  values  of  x,  y,  z,  iv'm  the  equations  of 
condition,  furnishes  a  most  thorough  proof  of  the  accuracy 
of  the  entire  computation. 


Weights  of  the  Most  Probable  Values  of  the  Unknown  Quantities. 

33.  In  case  of  a  single  unknown  quantity  determined  by 
direct  observation,  the  computation  of  the  weight  of  the 
arithmetical  mean  was  found  to  be  very  simple.  In  the  case 
under  consideration,  where  the  equations  to  be  solved  con- 
tain several  unknown  quantities,  the  difficulty  is  greatly 
augmented. 

In  our  equations  of  condition  we  have  supposed  the  quanti- 
ties observed  to  be  «,,  «„,  ws,  etc.  We  have  already  shown  that 
if  the  resulting  equations  of  condition  are  not  of  equal  weight, 
they  may  be  made  so  by  multiplying  each  by  the  square 
root  of  its  respective  weight.  We  shall  therefore  in  investi- 
gating the  weights  of  the  unknown  quantities  assume  the 
weight  of  each  observation  to  be  unity. 


§  33-  WEIGHTS  OF   UNKNOWN  QUANTITIES.  .55 

Let  px,  /„,  A,  A,,  be  the  weights  of  x,y,  z,  and  w  respectively  ; 

£x>  fy>  £z>  *w,  their  mean  errors. 
Let  £  be  the  mean  error  of  an  observation. 

As  all  of  our  equations  are  linear,  it  is  evident  that  if  the 
elimination  of  the  three  unknown  quantities  x,  y,  and  z  be 
completely  carried  out,  the  resulting  equation  will  give  w  as 
a  linear  function  of  nlt  n»  ;/3,  etc.  Similarly,  if  x,  y,  and  w  be 
eliminated,  we  shall  have  z  expressed  as  a  linear  function  of 
the  same  quantities,  and  so  of  each  of  the  others. 

We  may  therefore  write 

x  —  afa  +  a^  -f  a3n3  +  etc.; 

y  =  A»i  +  A*.  +  A».  +  etc.;  . 

*  =  X,»,  +  r^2  +  r3«3  +  etc.;  ' 
w  =  dfa  +  ^a«,  +  ^8ws  +  etc.; 

a',  /?,  etc.,  being  numerical  coefficients  and  functions  of  a,  b, 
etc. 

We  have  now  from  (31),  remembering  the  above  notation, 


x  —  £  Vet?  +  a?  +  a*  +  etc.  =  £  V\aa\. 


From  (33),        A  '=        =  .  .  .  p.  =         .          .    .     (66) 


The  weights  therefore  become  known  when  we  have  the 
values  of  \_aa~\  .  .  .  [tftfj.  For  this  purpose  we  must  make  use 
of  the  normal  equations  (41),  which  for  convenience  of  refer- 
ence are  here  rewritten  : 


56  •  LEAST  SQUARES.  §  33- 

•      (41) 
Let  us  now  assume  the  following  system  of  equations : 

•  (67) 


\aa~\x  -f-  \_aV\y  +  \_ac\e  +  [ad~\w  =  [an]  ; 
\ab~\x  +  \bb\y  +  [bc~]z  +  {bd~\w  =  \bn\  • 
\_ac\x  +  \bc-\y  +  \cc-\z  +  \_cd~\w  =  [«,]  ; 
[ad]x  +  \bd]y-\-  [cd~\z  +  \dd]w  =  \dii\  . 


\aa-\Q  +  \_aU\Q  +  \_ac~\Q"  +  \_ad}Q"  =  o  ; 
\_aU\Q  +  \bb\Q  +  \bc\Q"  +  \M-\Q"  =  o; 
\_ac\Q  +  \bc-\Q  +  \cc\Q'  +  \cd1Q"  =  o  ; 
\af\Q +\bd\Q  +  \cd\Q'  +  [dd}Q"'  =  i  • 


These  equations  will  be  possible,  as  there  are  four  unknown 
quantities,  Q,  Q',  Q' ',  and  Q",  and  four  equations  for  determin- 
ing their  values ;  further,  as  the  equations  are  of  the  first  de- 
gree there  will  only  be  one  system  of  values  for  Q,  Q',  etc. 

Now  let  the  normal  equations  be  multiplied  by  Q,  Q',  Q", 
and  Q"j  in  their  respective  orders,  and  the  resulting  equations 
added.  Then  in  consequence  of  (67)  in  the  resulting  equations 
the  coefficients  of  x,  y,  and  2  will  be  zero,  and  that  of  w  unity. 
Therefore  we  shall  have 

w  =  \an~\Q  +  \bn~\Q  +  [cn]Q'f  +  \dn\Q".      .     (68) 

We  shall  now  show  that  Q"  =  [tftf],  and  is  therefore  the 
reciprocal  of  the  weight  of  w. 

Let  us  expand  the  quantities  contained  in  the  brackets, 
equation  (68),  and  compare  the  results  with  the  last  of 
equations  (64).  We  thus  find  the  following  values  of  dlt  $„ 
etc.: 


(69) 


§  34-  WEIGHTS  OF   UNKNOWN  QUANTITIES.  $f 

Multiplying  each  of  these  by  its  a  and  then  adding,  then 
multiplying  each  by  its  b,  c,  and  d  successively  and  adding, 
we  have  by  (67)  the  following  equations  : 

«A  +  «.*.  +  «.*.  +  ...  =  [«*]=  o  ; 


+  'A  +  'A  +  •  •  •  =  [«n  =o  ; 
.  +  4*.  +  4*.  +  •  •  •  = 

Now  let  each  of  (69)  be  multiplied  by  its  $  and  the  results 
added.     Then  by  (70)  we  have 

*A  +  *A  +  *A  +  •  •  •  =  [**]  =  &"•    Q-  E.  D.  (71) 

The  solution  of  equations  (67)  therefore  determines  the 
weight  of  w.  In  a  precisely  similar  manner  the  weight  of 
each  of  the  unknown  quantities  may  be  determined.  Thus, 
to  determine  the  weight  of  x,  we  write  for  the  second  mem- 
ber of  the  first  of  (67)  unity  instead  of  zero,  and  write  zero 
for  the  absolute  term  of  each  remaining  equation.  The  re- 
sulting value  of  <2  will  be  the  reciprocal  of  the  weight  of  x. 

This  process  is  simple  enough  in  theory,  but  its  application 
is  laborious,  as  we  must  solve  equations  (67)  separately  for 
the  weight  of  each  unknown  quantity.  This  does  not  involve 
so  great  an  amount  of  labor  as  may  at  first  appear,  as  much 
of  the  computation  will  already  have  been  performed  in  the 
solution  of  the  normal  equations.  It  is  easy,  however,  to 
derive  a  process  which  will  generally  be  much  more  con- 
venient. It  is  as  follows  : 

34.  In  the  solution  of  equations  (41)  by  successive  substitu- 
tions we  found  for  the  final  equations  in  w  —  see  (56)  — 

\ddj\w  =  D&3]. 

We  shall  now  show  that  the  coefficient  \dd^\  =  ^777,  and 
is  therefore  the  weight  of  w. 


LEAST  SQUARES.  §  34. 

For  this  purpose  let  us  write  equations  (41)  as  follows  : 

\ad\x  +  \_ab\y  +  \ac~\z  +  [ad]w  —  [an]  =  A  ; 
\ab~\x  +  [bb]y  +  \bc\z  +  [bd]w  -  [bn]  =  £; 
[»  +  \bc\y  +  \cc\z  +  \cd\w  -  [en]  =  C\ 
\ad~\x  -t-  [&/]>>  +  [«/>  +  [dd}w  -  [dn]  =  D. 


Let  us  now  suppose  the  equations  solved  by  means  of  the 
auxiliaries  0,  Q  ',  Q",  and  Q'",  determined  from  (67),  when  we 
shall  have 

w  =  \an\Q  +  \bii\Q'  +  \cn\Q'  +  \dn\Q" 

(72) 


This  will  now  be  the  same  value  of  w  as  before  obtained,  if 
we  make  A=B=C=D  —  o. 

Let  us  now  suppose  the  equations  solved,  as  before,  by 
substitution.  Since  in  this  process  no  new  terms  in  D  are 
introduced,  the  coefficient  of  D  will  not  be  changed  in  the 
final  equation  for  w,  and  we  shall  have 

\dd$\w  =  [dn  3]  +     D      -f-  terms  in  A,  B,  and  C; 
from  which   w  =  j^Q  +  _^_  +  terms  in  A,  B,  and  C. 


Now  it  is  evident  that  the  coefficients  of  A,  B,  C,  and  D  must 
be  the  same  in  this  equation  as  in  the  value  before  obtained, 
equation  (72).  Therefore 

Q"  =—  —  .  Q.  E.  D. 


We  therefore  see  that  we  can  obtain  the  values  of  the  un- 
known quantities  from  equations  (41),  and  at  the  same  time 
their  respective  weights,  by  arranging  the  elimination  so  that 


§35-  WEIGHTS  OF   UNKNOWN  QUANTITIES.  S9 

each  in  succession  shall  come  out  last.    The  coefficient  of  the 
unknown  quantity  in  the  final  equation  will  be  its  weight. 

35.  In  solving  a  system  of  four  equations  like  the  above 
it  is  best  to  proceed  as  follows:  Let  w  be  determined,  as 
above,  by  substitution  in  the  order  x,  y,  2.  We  then  have 
w  with  its  weight  from 

\ddj\w  =\_dnj\. 

Equations  (56)  then  give  successively  z,  y,  and  x. 

Let  now  the  elimination  be  performed  in  the  opposite  order, 
viz.,  w,  z,y,  when  we  have  x  with  its  weight  from  the  equa- 
tion 

\aa  3]*  =  [an  3], 

\aa  3]  being  the  weight  of  x. 

This  value  of  x  must  agree  with  the  former  value  within 
the  limits  of  error  of  the  computation,  thus  furnishing  a  con- 
venient check  to  the  accuracy  of  the  computation. 

For  the  weight  of  y  and  z  we  need  not  repeat  the  elimina- 
tion, but  proceed  as  follows : 

Let  us  suppose  the  elimination  performed  in  the  order  x, 
y,  w,  z.  We  shall  then  have  the  same  auxiliary  coefficients 
as  in  the  first  case,  as  far  as  those  indicated  by  the  numerals 
i  and  2,  and  equations  (52)  will  be  the  same  as*  before;  but 
as  the  elimination  will  now  be  performed  in  the  order  w,  z^ 
instead  of  #,  w,  we  write  them 

\_dd2\w  +  \cd2\z  —  \dn2\  ; 
\cd2\w  +  \cc  2\z  =  \cn  2]  . 

From  the  first  of  these, 

\dn  2]        \cd  2] 

7£/    k i  t J^T 

\_dd2]          \dd2Y 


60  LEAST   SQUARES.  §  35- 

Substituting  this  in  the  second  gives  us  for  the  coefficient 
of  2 


But  we  have     \dd$\  F|  \ddz\  - 
From  these  two  equations  we  find 

\cc<\  -  \cc 
And  in  a  similar  manner, 


We  therefore  have  the  following  precepts  and  formulae 
for  computing  the  weights  in  the  case  of  four  normal  equa- 
tions : 

First,  perform  the  elimination  in  the  order  x,  y,  z,  w, 

• 

then    /„  =  [ddj] ; 

-(73) 
Second,  perform  the  elimination  in  the  order  w,  z,  yyx, 

then    px  =  \aa  3]  ; 


§  36.  WEIGHTS  GF   UNKNOWN  QUANTITIES.  6l 

The  formulas  for  the  auxiliary  coefficients  for  the  second 
elimination  may  be  derived  from  those  for  the  first  by  simply 
interchanging  the  letters  a  and  d  and  b  and  c.  The  process 
is  so  simple  that  it  will  be  unnecessary  to  write  them  out  in 
full. 


Other  Expressions  for  the  Weights. 

36.  When  the  equations  have  been  solved,  as  already  ex- 
plained, and  the  various  checks  applied,  so  that  the  computer 
is  convinced  that  the  results  obtained  are  reliable,  it  may  be 
undesirable  to  repeat  the  elimination  merely  for  determining 
the  weights  of  the  first  and  second  unknown  quantities.  We 
may  derive  convenient  expressions  for  computing  the  weights 
in  this  case,  as  follows  : 

Suppose  four  solutions  of  the  equations  to  be  carried 
through  so  that  each  unknown  quantity  in  turn  is  first  deter- 
mined, the  order  of  the  others  remaining  the  same  :  we  should 
then  have  each  unknown  quantity  with  its  weight  completely 
determined,  as  we  have  already  seen.  The  solution  of  the 
equations  for  which  we  have  given  the  complete  formulae  is 
in  the  order  d,  c,  b,  a,  where  we  have  written  the  coefficients 
instead  of  the  unknown  quantities.  If  now  we  substitute  the 
values  of  w,  z,  and  y  in  the  third,  second,  and  first  of  equations 
(56)  in  order,  we  have  finally  the  expression  for*,  which  will 
be  a  fraction  with  the  denominator 


\aa\\bbi\\cc2\\ddj\. 

In  the  four  solutions  which  we  have  supposed  made,  the  un 
known  quantities  last  determined  will  be  in  succession  x^x^x 


62  LEAST  SQUARES.       /  §  36- 

y,  and  the  denominators  of  the  expressions  for  their  values  will 
be  as  follows  : 

\_aa\d\bb  i 

\_aa\\bb\\\dd2\  [> 
\aa\\cc\\\dd2\  \bb 
\bb\\cc  i\ 


where  the  subscripts  show  which  unknown  quantity  is  first 
determined  in  each  solution.  As  the  elimination  is  performed 
by  successive  substitutions,  no  new  factors  being  introduced, 
it  follows  that  these  expressions  are  equal  to  each  other  re- 
spectively. 

It  is  evident  that  when  the  order  of  the  elimination  is 
changed  so  that  a  different  quantity  is  first  determined,  the 
order  of  the  others  remaining  the  same  as  before,  the  values 
of  the  auxiliary  coefficients  \bb  i],  \_cc2~],  etc.,  which  do  not 
contain  the  coefficient  of  this  quantity  will  remain  as  before. 

Suppose,  as  above,  the  unknown  quantities  to  be  determined 
in  the  order  d,  c,  b,  a.  Now  let  a  second  solution  be  made  in 
the  order  c,  d,  b,a\  then  all  of  the  auxiliary  coefficients  as 
far  as  those  designated  by  the  numerals  i  and  2  will  remain 
as  before.  In  a  third  solution  following  the  order  b,  d,  c,  a, 
the  coefficients  designated  by  the  numeral  i  will  have  the 
same  values  as  in  the  first  case  ;  while  in  a  fourth  determina- 
tion in  the  order  a,  d,  c,  b,  they  will  all  differ  from  the  first 
series  of  values. 

Thus  indicating  by  the  subscripts  only  those  coefficients 
which  have  values  different  from  those  given  by  the  first 
elimination,  we  have  the  following  equations  : 

\aa\  \bb\\  \cc2]  \dd$\  =  [aa]  [bb\\  \dd2\  [or  3]-, 
[aa]  \bb  i]  [cc2]  [_ddf[  =  [aa]  \cc  i]  \dd2 
\_ad\  \bb\~\  [cc2]  \ddi\  =  \bU\  \cci 


36.  WEIGHTS  OF   UNKNOWN  QUANTITIES, 


We  already  have  the  weight  of  w.     The  weights  of  z,  y,  and 
x  are  given  by  these  last  equations,  viz. : 


"    •    (74) 


In  applying  these  formulae  the  following  additional  auxiliary 
coefficients  must  be  computed  : 


[cc  i],  =  \ee\         -         [ir]  ; 


[M] 


=  \_ddll 


\tci\ 


•    •    •    (75) 


In  case  of  three  unknown  quantities  the  formulae  become 


A  = 


(76) 


where  [^  i]0  has  the  value  given  above. 


64 


LEAST  SQUARES. 


§37- 


37.  An  elegant  expression  for  the  weights  is  obtained  by 
making  use  of  the  determinant  notation.  Thus,  referring  to 
the  normal  equations  (41), 


6S\    [bc 

[«*J+ 


aa\  [at 


ill 


M 


ad' 


2r//,  the  reciprocal  of  the  weight  of  w,  given  by  equations  (67), 
is  the  same  as  the  value  of  w  obtained  from  the  above  equa- 
tion by  making  [an]  =  \bn\  =  [en]  =  o  and  \dri\  =  I. 

Therefore  writing  A  for  the  complete  determinant  which 
forms  the  denominator  of  the  above  expression,  D"f  for  the 
partial  determinant  formed  by  dropping  the  last  horizontal 
line  and  last  vertical  column,  D"  for  the  partial  determi- 
nant formed  by  dropping  the  third  horizontal  line  and  third 
vertical  column,  and  similarly  D'  and  D  fpr  the  other  two, 
we  have 


/„  = 


D> 


A  =  TTM 


A_ 
~D"'' 
A 
D' 


(77) 


A  number  of  other  forms  may  be  derived  for  the  weights, 
all  of  which  involve  about  the  same  numerical  operations  as 
the  above.  In  certain  special  cases  different  forms  may  be 
more  convenient,  but  for  our  immediate  purposes  it  will  not 
be  necessary  to  develop  the  subject  further. 

It  may  readily  be  seen  from  what  precedes  that  the  rela- 
tive weights  of  the  unknown  quantities  may  be  derived,  even 
when  the  number  of  observations  does  not  exceed  the  num- 
ber of  unknown  quantities.  No  probable  errors,  however, 
can  be  determined  in  this  case. 


§  38.         MEAN  ERRORS   OF   UNKNOWN  QUANTITIES.  65 

Mean  Errors  of  the  Unknown  Quantities. 

38.  For  determining  the  mean  and  probable  error  of  an 
unknown  quantity  nothing  further  is  required  except  the  ex- 
pression for  the  mean  error  of  an  observation.  It  is  supposed 
that  the  equations  of  condition  have  been  reduced  to  the 
common  unit  of  weight  by  multiplying  each  equation  when 
necessary  by  the  square  root  of  its  weight. 

The  values  of  x.  j/,  z,  and  w,  as  deduced  above,  are  the  most 
probable  values  as  deduced  from  the  given  data.  When 
substituted  in  the  equations  of  condition  the  residuals 
^i»  v»  vv  etc.,  will  not  be  the  true  errors  unless  the  derived 
values  x,  y,  z,  and  w  are  absolutely  the  true  values,  a  condi- 
tion not  likely  to  be  realized. 


Let  (x  -\-  dx),  (y  -f-  <?/),  (z  -\-  6z),  (w  +  6w)  be  the  true  values  ; 
Alt  A^  ^3,  ...  Am,  the  true  errors. 

We  shall  then  have  two  systems  of  equations,  as  follows : 

•    •    (78) 


"(79) 


Let  us  multiply  each  of  equations  (78)  by  its  v  and  add  the 
resulting  equations.  Then  by  (40)  the  coefficients  of  x,  y,  z, 
and  w  will  vanish,  giving  us  the  relation  before  derived, 

\yn\  =  \yv\ (80) 


66  LEAST  SQUARES.  §  38. 

Proceeding  in  the  same  manner  with  (79),  we  find 

\vn\  =  \vA-}  ........     (81) 

Therefore  \vA~\  =  [vv]  ........    (82) 

In  order  to  obtain  an  expression  for  the  sum  of  the  squares 
of  the  true  errors,  viz.,  \_AA\,  in  terms  of  the  sum  of  the 
squares  of  the  residuals  [vv],  let  us  first  multiply  each  of 
equations  (78)  by  its  A  and  add  the  resulting  equations  ; 
secondly,  let  us  multiply  each  of  (79)  by  its  A  and  add  in 
like  manner.  The  results  are  as  follows  : 


\aA\x  +  \bA}y  +  \cA\z  +  \dA\w  -  \nA\  =  -  \vA\  =  -  [vv\  ; 
\aA-\  (x  +  dx)  +  \bA\  (y+6y)  +  \cA\  (z  +  d?) 

+  \dA\  (w  +  dw)  —  [tiA]  =  —  {A  A"}. 

Subtracting  the  first  of  these  from  the  second,  we  obtain 
\AA~\  =  [vv]  —  \_aA\Sx  —  \bA~\8y  —  \cA~\dz  —  \dA~\8w.  (83) 

If  we  could  now  assume  8x,  dy,  dz,  and  Sw  to  vanish,  we 
should  obtain,  since  ms2  =  \AA\  by  definition, 


m 


This  will  give  us  a  close  approximation  to  the  true  value  of 
£  when  m  is  large. 

For  a  more  accurate  determination  of  £  we  must  endeavor 
to  find  approximate  values  of  [aA~\dx,  \_bA\8y,  etc.  The  true 
values  are  beyond  our  reach,  but  principles  already  estab- 
lished give  us  a  means  of  approximation. 

Multiplying  each  of  equations  (79)  by  its  a,  and  adding, 
we  have 

\_ad\x    +  \_ab~\y  +  \ac~\z    +  \_aeT\w  —  [an]  j  _  __  r  ^ 
+  \ad\8x  +  \aU\$y  +  \ac\8z  +  [ad]dw  \ 


§38-         MEAN  ERRORS   OF   UNKNOWN  QUANTITIES. 


67 


Comparing  this  with  (41),  we  see  that  the  first  line  is  equal 
to  zero. 

Multiplying  each  equation  of  (79)  by  its  b  and  adding, 
then  in  a  similar  manner  by  its  c  and  </and  adding,  we  have 
finally 


\ad\8x  +  \ab\dy  +  \_ac~\K  z  +  \ad~\8w  =  —  \ad]  ; 
\ab\8x  -|-  \bb\$y  -\-  \bc~\8z  +  \bd~\8w  =  —  \bA\ ; 
\ac\dx  +  [^]<5>  +  \cc~\8z  +  [^Jdze;  =  -  \cA\ ; 


(34) 


Comparing  these  with  (41),  we  see  that  they  are  of  precisely 
the  same  form,  the  unknown  quantities  being  in  this  case 
dx,  dy,  dz,  and-^ze/,  instead  of  x,  y,  z,  and  w,  and  the  absolute 
terms  having  —  A  in  the  place  of  n.  The  solution  will  there- 
fore have  the  form — see  (64) — 


.    .    (85) 


If  we  now  write   these  values   in  (83),  we   shall   have  for 
j  etc.,  the  following  values: 


\bS\Sy  = 
\cA~\Sz  = 
\_dA-\Sw  =  (dA  +  d,A,  +  d,A,  +  ...) 


K86) 


In  regard  to  these  products  it  is  to  be  remarked  that  they 
must  necessarily  be  positive,  as  our  conditions  require  \vv] 


68  LEAST  SQUARES.  §  38. 

to  be  a  minimum.  Any  system  of  values  of  x,y,  z*  and  w,  there- 
fore, differing  from  those  derived  from  the  normal  equations 
(41)  must  increase  the  sum  of  the  squares  of  the  residuals. 
Therefore  \AA\  >  \yv\,  and  the  terms  following  \vv\  in  (83) 
must  be  positive. 

Let  us  now  perform  the  indicated  multiplication  in  (86). 

Confining  ourselves  to  the  last  equation,  since  the  form  is 
the  same  for  all,  we  can  indicate  the  result  as  follows  : 

-  \dA-\dw  =  </AAA+  4A44+  4A44+  •  •  •  +  ^>K4A). 

The  last  term  indicates  the  sum  of  all  the  terms  formed  by 
multiplying  together  different  values  of  ^,  as  AA>  ^,A»  •  •  • 
^m-i^m*  Now,  since  positive  and  negative  errors  occur 
with  equal  frequency  when  the  number  of  equations  of  con- 
dition is  very  large,  we  may  assume  this  term  equal  to  zero.* 
Writing  for  (^,^,),  (^2A)>  etc.,  the  mean  value  of  those 
quantities,  viz.,  £2,  and  placing  for  [dd]  its  value  from  the  last 
of  (70),  viz.,  [dd]  =  i,  we  have 


In  a  manner  precisely  similar  we  find 

-  \aA~\dx  —  -  \bA]6y  =  -  \cA~\S 
Therefore  equation  (83)  becomes 

m?  =  \w\  +  45*. 
From  which  f== 


In  this  case  there  are  four  unknown  quantities.     In  general 
if  the  number  of  unknown  quantities  is  //,  we  shall  have 


/  M" 
=  ±  y  J^TT;- 


(88) 


*  Also  positive  and  negative  values  of  d^ ,  d*  .  .  . ,  dt  ,  £a  .  .  .  are  equally  pro 
able. 


§3§.         MEAN  ERRORS   OF   UNKNOWN  QUANTITIES.  69 

With  the  values  of  px,py,pzy  and  pw  computed  by  (73),  we 
have  finally  » 

•„=-£=;  (89) 


and  the  probable  errors  of  x,  y,  z,  and  w  will  be  obtained  by 
multiplying"  these  respectively  by  .6745. 

We  have  now  developed  the  subject  as  far  as  is  necessary 
for  our  purposes.  A  complete  example  of  the  solution  of  a 
series  of  equations  with  three  unknown  quantities,  together 
with  the  determination  of  their  respective  weights  and 
probable  errors,  will  be  found  in  connection  with  article 
(191)  of  this  volume. 


INTERPOLATION. 


39.  In  the  Nautical  Almanac  are  given  various  quantities, 
such  as  the  right  ascension  and  declination  of  the  sun,  moon, 
and  planets,  places  of  fixed  stars,  etc.,  which  are  functions  of 
the  time.  This  is  assumed  as  the  independent  variable,  or 
argument  as  it  is  termed  by  astronomers.  The  ephemeris 
gives  a  series  of  values  of  the  function  corresponding  to 
equidistant  values  of  the  argument.  In  case  of  the  moon, 
which  moves  rapidly,  the  position  is  given  at  intervals  of  one 
hour;  the  place  of  the  sun  is  given  at  intervals  of  twenty -four 
hours ;  while  the  apparent  places  of  the  fixed  stars  vary  so 
slowly  that  ten-day  intervals  are  sufficiently  small.  When 
any  of  these  quantities  are  required  for  a  given  time,  this 
time  will  generally  fall  between  two  of  the  dates  of  the  ephe- 
meris— seldom  coinciding  with  one  of  them  ;  the  required 
value  must  then  be  found  by  interpolation. 

Interpolation  in  general  is  the  process  by  ivhich,  having  given 
a  series  of  numerical  values  of  any  function  of  a  quantity  (or  argu- 
ment],  the  value  of  the  function  for  any  other  value  of  the  drgii- 
ment  may  be  deduced  without  knowing  the  analytical  form  of  the 
function. 

We  shall  consider  the  subject  more  in  detail  than  will  be 
necessary  for  the  simple  purpose  of  using  the  ephemeris, 
on  account  of  its  importance  in  other  directions. 

In  what  follows  we  shall  suppose  the  values  of  the  function 
given  for  equidistant  values  of  the  argument,  which  will 
always  be  the  case  practically.  Also  the  intervals  must  be 


§39  INTERPOLATION,   GENERAL  FORMULA.  7  I 

small  enough,  so  that  the  function  will  be  continuous  between 
consecutive  values  of  the  argument. 

Let  w  —  the  interval  of  the  argument. 

.....(7--3H  (T-2w\  (T-w\  (T),  (T+w),  (T+2w\ 
(T-\-$w),  .  .  .  =  the  values  of  the  argument. 

The  notation  for  the  arguments,  functions,  and  successive 
differences  will  be  shown  by  the  following  scheme  : 

Argu-  ist  ad  3d  4th  5th 

ment.    Function.     Difference.     Difference.      Difference       Difference.     Difference. 

)w} 

w)J  ~(7~£^)  f"(T-2w)      , 

W)  ''<7--t»>;,.        ,  f  (T-^f-(T- 
}  " 


The  notation  shows  at  once  where  each  quantity  belongs 
in  the  scheme.  The  first  differences  are  formed  by  subtract- 
ing each  function  from  the  quantity  immediately  following 
it,  the  argument  being  the  arithmetical  mean  of  the  arguments 
of  the  two  functions.  Similarly  the  second  differences  are 
formed  by  subtracting  each  quantity  in  the  column  of  first 
differences  from  the  one  immediately  below  it,  and  so  on  for 
the  successive  orders  of  differences.  It  will  be  observed  that 
the  even  orders  of  differences,  f"ifiv,  etc.,  fall  in  the  same 
horizontal  lines  with  the  functions  themselves,  and  have  the 
same  arguments,  while  the  odd  orders,  /"',  f",  etc.,  fall  be- 
tween those  lines.  The  even  differences  all  have  integral  argu- 
ments, and  the  odd  differences  fractional  arguments. 

The  arithmetical  mean  of  two  consecutive  differences  is 
indicated  by  writing  it  as  a  function  of  the  intermediate 
argument.  For  example  : 

/"(  T)  =  ![/*(  T-&)    +  /•(  T  +  i«0]  ; 

a,)]. 


72  IXTERPOLA  TION,  §  40. 

40.  Suppose  now  we  set  out  from  the  function  whose  argu- 
ment is  T.     Evidently, 


2f'(T+  iX>  +f"(T  +  w)  ; 
f(T+  310)  =  /(T+  2w)  +f'(T+  » 

=f(T)  +  3f'(7+  fr>)  +3/"(  T+w)+f'"(  T+  j 


Proceeding  in  this  manner,  we  readily  discover  the  law  of 
the  series;  viz.,  the  coefficients  are  those  of  the  binomial 
formula,  and  each  successive  function,  f',f",  etc.,  is  on  the 
horizontal  line  drawn  under  the  one  which  immediately  pre- 
cedes*it.  Thus  we  have  the  general  formula 


f(T+  nw)  =f(T) 


+  •  •  (90 

If  we  assign  integral  values  to  n  we  obtain  the  tabular 
values,  viz.,/(7"-|-  w),f(T-{-  2w\  etc.;  but  the  formula  is  not 
used  for  this  purpose,  but  for  interpolating  between  the 
tabular  values,  in  which  case  n  is  fractional  and  must  be  ex- 
pressed in  terms  of  the  interval  of  argument  w  as  the  unit. 

41.  A  more  convenient  form  may  be  given  to  this  expres- 
sion (91),  as  follows  :  We  have 


)=  /'"(  T+  &)  +/'"(  T)  +/"(  T+  *W)  ; 
*•  (7-+  2w)  =/'"(T)+  2f(T+  *w)  +/«(T)  +/""(  T+ 


§41-  INTERPOLATION,   GENERAL   FORMULA,  73 

Substituting  these  values  in  (91)  and  reducing,  we  readily 
obtain 


f(T+  nw}  =/(T)  +  nf(T+  fc*) 
«+      »<*  - 


/'"(  7-+  jw) 


The  law  of  the  series  is  obvious  ;  viz.,  a  factor  is  added  to  the 
numerator  of  each  succeeding  coefficient  alternately  after 
and  before  the  other  factors,  the  last  factor  of  the  denomi- 
nator being  the  same  as  the  order  of  differences.  The  succes- 
sive differences  are  taken  alternately  below  and  above  the 
horizontal  line  drawn  immediately  below  the  function  from 
which  we  set  out. 

Formula  (92)  will  be  used  for  interpolating  forward.  For 
interpolating  backward  a  better  form  may  be  derived  by 
writing  for//(T+  %i»\f"'(  T  +  %w),  .  .  .  their  values  in  terms 
oif'(T  -  %w),f"\T  -  \w\  .  .  .  viz.  : 


Changing  n  at  the  same  time  into  —  #,  since  the  formula  is 
to  be  used  for  interpolating  backwards,  we  readily  find 


f(T-nw)=f(T}-nj 

(n  -f-  i)  n  (n 


1.2.3 

(n+i)n(n-iY,n-2)  ,    } 

1.2.3.4 


74  INTERPOLA  TION.  §  42. 

42.  In  applying-  (92)  and  (93)  it  will  be  more  convenient 
to  write  them  as  follows  : 


.     .     (92)l 


AT-  nw)  =  f(T)  -  n 


.    -     (93), 


In  ($2\  and  (93)!  each  difference  is  used  to  correct  the  one  of 
the  next  lower  order  immediately  preceding  it,  and  the  quanti- 
ties to  be  multiplied  will  generally  be  small.  In  interpolating 
a  value  of  the  function  corresponding  to  a  value  of  the  argu- 
ment between  Tand  (7"+  %w),  we  use  (92),  and  set  out  from 
f(T).  If  the  argument  is  between  (T  +  %w)  and  (T  -}-  w), 
we  use  (93),  and  set  out  from  f(T-\-  w). 

When  the  interpolation  is  carried  to  any  given  order  of 
differences,  as  the  fifth,  it  is  a  little  more  accurate  to  take  the 
Arithmetical  mean  of  the  last  differences,  which  fall  immedi- 
ately above  and  below  the  horizontal  line  drawn  in  the  vicinity 
of  the  required  function.  Thus  the  last  term  of  (92),  and 
(93),  would  be  fl(T). 

43.  For  the  quantities  tabulated  in  the  American  Ephe- 
meris  it  will  only  be  necessary  to  carry  the  interpolation  to 
second  differences  ;  but  for  computing  ephemerides  or  tables 


§44-  INTERPOLATION,    EXAMPLE.  75 

of  any  continuous  function,  much  labor  is  saved  by  comput- 
ing the  quantity  directly  for  a  comparatively  few  dates  and 
supplying  the  intermediate  values  by  interpolation.  If  the 
function  is  of  such  a  character  that  some  order  of  differences, 
as  the  third,  fourth,  or  any  other,  vanishes,  this  gives  exact 
values  for  the  interpolated  quantities,  and  in  fact  the  process 
may  then  be  used  for  computing  values  of  the  function  for 
any  value  whatever  of  the  argument.  It  is  on  this  principle 
that  "tabulating  engines"  are  constructed. 

44.  As  an  example  of  the  application  of  (90),  (92),,  and  (93),, 
we  take  from  the  American  Ephemeris  the  following  values 
of  the  moon's  right  ascension  for  intervals  of  12  hours: 

1883, 

July         h/=«-  /'•  /"•       /'"•       /*•       /*• 

3d,     oh  5.45  15.68 

29  39.05 
i2h  6  14  54.73  —  27.08 

29  11.97  —  6.91 

4th,    oh  6  44    6.70  —  33.99  +  2.01 

28  37.98  —  4.90  —  .06 

I2h    7    12    44.68  -  38.89  +  1.95 

27  59.09  -  2.95  -  .01 

5th,   oh  7  40  4377  -  41-84  +  1.94 

27  17.25  —  i.oi  —  .16 

I2h  8     8     1.02  -42.85  +  1.78 

26  34.40  +  .77  -  .33 

6th,    oh  8  34  35.42  —  42.08  +  1.45 

25    52.32  +2.22  -.33 

I2h   9      O   27.74  —  39-86  +  1. 12 

25   12.46  +3-34 

7th,   oh  9  25  40.20  -  36.52 

24  35-94 
i2h  9  50  16.14 


76  INTER  POL  A  TION.  §  44- 

Example  i.     As  an  example  of  the  application  of  (92),,  let 
us  interpolate  the  moon's  right  ascension  for  1883,  July  5th, 


4h- 


Since  the  interval  of  the  argument  w  is  here  I2h,  we  have 
in  this  case  nw  =  4h,  or  n  —  T%  =  i-  Setting  out  from  July 
5th,  oh,  we  have 


w]  .01 

f\T  +  \w)       =  -      .16  .-.  f\T)  =  -  .085 


-y- fv         =  ~       -040 

fiv        =  +     L94Q 

Corrected, /iv       =  +    1.900 

~,       /•»  i 

/*+...=  - 


4 

f"       —  —    i.oio 


Corrected,  f"        =  —     1.802 

^  {/"'+-..=  -      -80. 

/"  -41.840 

Corrected,/"  -  42.641 

^-r  ]/"+...=  + 14.214 


Corrected,/'        =27m3i8. 
/  =  a  =  7"4o"'43'-77 


1883,  July  5th,  4h,  a  ==  7h49ni54s.26 

This  value  agrees  exactly  with  that  found  in  the  American 
Ephemeris  for  1883  (see  page  115). 


§44-  INTERPOLATION,   EXAMPLE.  77 

Example  2.  Let  us  now  apply  (93^  to  determine  the  moon's 
right  ascension,  July  5th,  2Oh.  Here  we  set  out  from  July  6. 
As  before,  n  —  \,fv(T]  —  —  .33. 

n  -\-  2 

^j-r    =+  .154 
fiv    =  +  1450 

Corrected,  fiv       =  -\-    1.604 
n~2  «  fiv_  66g 

4     r 

/'"       =+__77o 
Corrected,  /'"       =  -f 


I  J      r/lf  x- 

— —    |/  -...—     -  .639 

/7/  —     —    42.080 

Corrected,  f"  -  42.719 

-  -          \f"  -...=  -  14.240 


Corrected,/'         =; 


/=«  = 
1883,  July  5th,  20h  a  — 

The  algebraic  signs  of  the  various  corrections  are  deter- 
mined without  difficulty,  as  follows:  If  a  horizontal  line  be 
drawn  in  the  table  of  functions  and  differences  (p.  75)  in  the 
vicinity  of  the  given  argument  (in  the  first  of  the  above 
examples  immediately  below  5doh),  the  successive  differences 
required  will  fall  alternately  below  and  above  this  line. 


/  8  INTERPOLATION.  §45. 

Beginning  with/r  we  determine  the  correction  tofiv,  which 
is  to  be  applied  so  as  to  bring  the  value  nearer  to  that  imme- 
diately below  the  line.  In  this  case/'1'  =  -f-  1.94;  that  which 
immediately  follows  is  +  1.78  ;  therefore  the  correction  must 
be  subtracted  from  1.94,  giving  the  corrected  fiv  =  1.90. 

The  value  of  f"  is  —  i.oi  ;  the  value  immediately  above 
the  line  is  —  2.95.  The  first  must  be  corrected  so  as  to 
bring  it  nearer  the  latter,  giving  in  this  case  the  corrected 
f"  —  —  1.802,  and  so  on  for  each  difference  in  succession. 
That  is, 

When  the  quantity  is  <  !•  the  horizontal  line,  apply 

the  correction  so  as  to  bring  it  in  the  direction  of  the  one  in 


the  same  vertical  column  immediately  j  ^eJJ^  [  it 


Special  Cases. 

45.  Whenever  (92),  or  (93^  can  be  applied,  nothing  more 
will  be  necessary  ;  they  require,  however,  a  knowledge  of 
the  value  of  the  function  for  several  dates  both  before  and 
after  those  between  which  the  interpolation  is  made.     It  is 
sometimes  necessary  to  interpolate  between  values  of   the 
function  near  the  beginning  or  end  of  the  table  :  as,  for  in- 
stance, we  might   require  from  the  tabular  values   of   the 
moon's  right  ascension,  given  on  page  75"  to  determine  the 
value  between  the  dates  July  3d,  oh,  and  3d,^i2h,  or  between 
7th,  oh,  and  7th,  I2h.     In  either  of  these  cases  the  series  of 
differences  terminates  with  f'\    so    the  above  formulae  will 
only  give  the  value  to  first  differences  inclusive. 

We  shall  consider  the  two  cases  separately. 

46.  First.     For  arguments  near  the  beginning  of  the  table. 
As  before,  calling  the  arguments  between  which  it  is  re- 

quired to  interpolate  the  function,  T  and   T  -\-  w,  we  may 
apply  formula  (91),  setting  out  iromf(T). 


§46.  INTERPOLATION,    SPECIAL    CASES.  79 

If  the  argument  for  which  the  value  of  the  function  is  re- 
quired is  nearer  T-\-w  than  T,  it  will  be  a  little  simpler  to 
set  out  from  T-\-  w  and  interpolate  backwards.  In  this  case 
the  formula  requires  the  following  modification: 

Changing  n  into  —  n,  we  have 


f(T-nw)=f(T)-nf'(T  +  &,)  +  f»(T  +  w) 


n(n  +i)(»  +  2)  (»  +  3)  Q  +  4) 
1.2.3.4.5 

From  the  manner  of  forming  the  successive  functions,  we 

have 


'  (  T  +  w  =/'(  r- 


'"  (  7-  +  f  «o  =  /"'<  r+iw)+/^(  7-  + 

-  (  T  +  aw)  =  /*(  T  + 


Substituting  these  values  in  the  above  and  reducing,  we 
have 


-  nf'(T- 


(«  -  i)n  (n  +i)(n  +  2) 

1.2.3.4  J    ^ 


I-2.3-4-5 


* 


80        *  INTER  POL  A  TION.  §  46. 

I 

For  greater  convenience  in  the  application,  (91)  and  (941 
may  now  be  written  as  follows  : 


.      .     (95 


n  —  I 


f(  T  -  nw)  =A  T)  +  n     -/'( T  -  *w)  +  —^    /"( T) 


.    -    (95), 


Example  3.  Required  the  moon's  right  ascension,  1883, 
July  3d,  4h.  Referring  to  the  series  of  values  (Art.  44),  we 
have  for  this  case  nw  =  4h  ;  .*.  n  —  ^. 

fv    =  -        .06 
'—r-. fv    =  +      -°44 

fiv    —  +     2.010 

Corrected,/**   =  +    2.054 

n  —  *  \   . 


4     .r 

f"  =  —    6.91 


.  'Corrected,  f"  =  -      8.279 


§46.  INTERPOLATION,    SPECIAL   CASES.  8l 


—-  !/'"...=  +    4-599 

£        /"  =  ^_2£o8_ 
Corrected,/"    =  —  22.481 


'  ' . . .  =  +    7.494 

f    =29™398.o5o 
Corrected,  /'    =29m468.544 

/=  a  =  5h45mi5s.68o 
1883,  July  3d,  4h,  a  =  5h55mns.i95 

Example  4.  Required  the  moon's  right  ascension,  1883, 
July  3d,  8h.  In  this  case  we  use  formula  (95)^  since  the 
argument  is  nearer  I2h  than  oh.  n  =  -J. 

-  /-  =  +   .06 


=  2.01 


Corrected,  /*w   =  +    2.05 


-  /'"  =  +_  6.910 

Corrected,  /'"  =  +    8.082 


//X     =     -    27.080 

Corrected,  f"  —  —  23.488 


82  INTERPOLA  TION.  §  47- 

n  —  I 


2   (/"•••  =  +  7.829 
-/'=- 


Corrected,/'  =  —  29m3is.22i 

*{--/'...==:--  9"'5o.s407 

f=  a  =  6hi4m548.730 

1883,  July  3d,  8h,  a  =  6h  5m  4^.323 

47.   Second.     Arguments  near  the  end  of  the  table. 
Proceeding  in  a  manner  precisely  similar  to  that  of  the 
previous  article,  we  readily  obtain  the  formulae 


(»_!)„(„-!-   l)(«  +  2)  _ 

1.2.3.4 

(»-l)»(»+l)(»  +  2)(*+3^(7._          }>  (      } 
1.2.3.4.5 


-  nw)  =  AT)  -  nf'(T  - 


1.2.3 


n(n_--_J.)  (n  -  2)  (n  -  $      ,     _       . 
1.2.3.4 

_  „(„-!)(„-  2)  („-  3)  (K-  4}^  ( 

1.2.3.4.5 

The  1        st  d  1  of  these  applies  for  interpolating  in  the 


47-  INTERPOLATION,    SPECIAL    CASES.  83 


direction  in  which  the  argument  j  ^^s^s  1  •      The  above 
may  be  written  as  follows  : 

f(T  +  nw)  =f(T)  +  »    f'(T  +  fr)  +  ^= 


f(T—  w) 
\  \      •    (98) 

f(T—nw)=f(T)-\-n\    -f'(T—^w)-\--      -\f"(T—w) 

v  2         (. 

^f*(T-2W) 

•        (98.) 


Example  5.     Required  the  moon's  right  ascension,   1883, 
July  ;th,  4h. 

»  =  $;         /u--.33;          /iv-  +  i.i2;          //7/-  +  3.34; 
f"  =  -  36.52  ;       f  =  24  35.94  ;       /=  9h25m4os.2o. 

Substituting  in  (98)  as  above,  we  find 


Example  6.  Required  the  moon's  right  ascension,  1883, 
July  7th,  8h. 

By  substituting  the  numerical  values  in  formula  (98),  we 
find  for  this  case 


It  will  be  observed  that  in  the  application  of  formulas  (95), 
(95)i»  (9^)'  and  (98),  the  algebraic  signs  of  the  various  correc- 


&4  INTERPOLA  TION.  §  48. 

tions  may  be  determined  in  a  manner  entirely  similar  to  that 
explained  in  connection  with  formulae  (92)  t  and  (93)^  (See 
Art.  44.). 

• 
Interpolation  into  the  Middle. 

48.  When  the  function  is  to  be  interpolated  for  a  value  of 
the  argument  half  way  between  two  consecutive  dates  of  the 
table,  this  is  called  interpolation  into  the  middle. 

For  this  case  either  (92^  or  (93),  may  be  used,  but  a  more 
convenient  formula  is  obtained  as  follows.  Write  £  in  place 
of  n  in  (92)  : 


Then  in  (93)  let  n  =  \,  and  set  out  from  (T  -\-  w)  : 


Taking  the  mean  of  these  equations,  observing  in  the  result- 
ing equation  that  the  coefficients  of  the  odd  differences, 
/',  /'",  etc.,  vanish,  and  writing 


=f"(T+ 


49-  PROOF  OF  COMPUTA  TION.  85 

lw) 

(99) 


or 


*)«.-4W          199), 

Example  7.  Let  it  be  required  to  determine  the  moon's 
right  ascension,  1883,  July  5th,  6h.  We  must  interpolate  into 
the  middle  between  July  5th,  oh,  and  July  5th,  12''. 

/*»  =  +    i.  860 

-  &fiv  =  -        -349 

f"  =  -  42.345 

Corrected,  f"  =  —  42.694 

-*•!/"..-  =  +    5-337 


Therefore  1883,  July  5th,6h,tf=7h54m278.73 

Proof  of  Computation. 

49.  The  method  of  differences  furnishes  a  very  convenient 
check  on  the  accuracy  of  a  computation,  when,  for  a  series 
of  values  of  an  argument  succeeding  each  other  at  regular 
intervals,  a  series  of  values  of  any  function  have  been  com- 
puted. Suppose  an  erroneous  value  of  one  of  these  quanti- 
ties, f(T)  -f-  x,  has  been  obtained,  x  being  the  error.  The 
functions,  with  the  respective  differences,  would  then  be  as 
follows  : 


-  4, 


86  INTERPOLATION.  §  50. 

Thus  the  error  x  in  the  function  has  increased  to  6x  in  the 
fourth  difference,  the  greatest  deviation  being  in  the  horizon- 
tal line  where  the  erroneous  value  of  the  function  is  found. 

Suppose,  for  example,  an  error  of  5s  had  been  made  in 
computing  one  of  the  values  of  the  moon's  right  ascension 
given  in  Art.  44.  The  scheme  of  differences  would  then 
be  as  follows: 

July  /  =  a  f  f"  f"  fiv 

h.    m.        s. 

3d,       oh       5  45  15.68 

29  39.05 
I2h      6  14  54.73  —  27.08 

4th,      oh 

I2h 

5th,     oh 


6th,     oh 

We  see  at  once  without  going  further  than  second  differ- 
ences that  the  value  for  July  4th,  I2h,  is  erroneous. 

Differential  Coefficients. 

50.  When  we  have  a  series  of  numerical  values  of  a  func- 
tion, corresponding  to  equidistant  values  of  the  argument, 
we  may  compute  the  numerical  values  of  the  differential  co- 
efficients from  the  tabular  differences  as  follows:  Either 
form  of  the  interpolation  formula  is  arranged  according  to 
ascending  powers  of  n.  The  function  f(T -\-  nw)  expanded 
by  Taylor's  formula,  and  the  differential  coefficients,  com- 
pared with  the  coefficients  of  the  different  powers  of  n  in  the 
above  expansions,  give  at  once  values  of  these  quantities. 


29 

II 

•97 

—  i 

.91 

6 

44 

6. 

70 

—  28 

•99 

-  17-99 

28 

42 

.98         —  lo.oo 

7 

12 

49-68 

-48 

.89 

+  31-95 

27 

54 

.09 

+  12 

.05 

7 

40 

43- 

77 

-  36-84 

—  1  8.06 

27 

17 

.25 

—  6.01 

8 

8 

i. 

02 

-42 

.85 

26 

34 

.40 

8 

34 

35- 

42 

§  5°-  DIFFERENTIAL    COEFFICIENTS.  87 

The  most  rapid  convergence,  and  consequently  the  best 
formulas,  will  be  obtained  by  introducing  into  formula  (92) 
the  arithmetical  means  of  the  odd  differences  situated  above 
and  below  the  horizontal  line  drawn  through  the  function 
irom  which  we  set  out,  using  the  notation  for  the  arithmeti- 
cal mean  given  on  page  71. 

From  the  manner  of  forming  the  differences  we  readily  see 

/'  (7-+  kw)  -=  f  (T)  +  if"(T)  ; 
f'"(T  +  J*0  =f 


These  values  being  substituted  in  (92),  we  readily  derive 

AT  +  nw)  =AT)  +  *f'(T)     +  —/" 


.  -  ~ 

1.2.3  1-2.3.4 

(n  +  2)  (n  +  i)  n  (n  -  I)  («  -  2) 
1.2.3.4.5 

Arranging  this  according  to  ascending  powers  of  n,  it  be- 
comes 


+[/'"(  T)  .  .  .]- 


.2.3.4.5.6' 


88  INTERPOLATION.  §51. 

Expanding  the  function  by  Taylor's  formula, 

df  d'f  n*w>      d'f  n'w' 


•  •  • 


Comparing  the  coefficients  of  like  powers  of  n  in  these  two 
series,  we  have  the  following  values  for  the  differential  co- 
efficients : 


f(T+  nw)  = 

d'f     rcV          d'f 


~* 


51.  Formulae  (ioi)  will  not  apply  to  values  of  the  function 
near  the  beginning  or  end  of  the  table.  We  obtain  formulas 
for  these  special  cases  by  comparing  formulas  (91)  and  (97), 
respectively — arranged  according  to  ascending  powers  of  n — 
with  Taylor's  formula.  We  thus  obtain — 

For  arguments  near  the  beginning  of  table  : 


-  */"(  T+w 


5L  DIFFERENTIAL    COEFFICIENTS. 

For  arguments  near  end  of  table  : 


Example  8.  Let  it  be  required  to  compute  the  numerical 
values  of  the  differential  coefficients  of  the  moon's  right 

da  d*oL 
ascension  with  respect  to  the  time,  -j~  -^Tr^  •    •    •  f°r  J883, 

July  5th,  oh. 

In  substituting  the  numerical  values  in  (ioi),  w,f',f"  .  ,  . 
must  all  be  expressed  in  the  same  unit.  It  will  be  convenient 
to  express  them  in  seconds. 

From  the  numerical  values  given  on  page  75  we  have 


—    1 .08 

j-T^rV^;  =  -  -000  °453 ; 


=  ~  -000  °°20- 

Therefore  ,  =  +  .038391  ; 

~  ~  •°°°972- 


This  value  of  -7=,  may  be  regarded  as  the  fractional  part  of 


90  INTERPOLATION.  §  52. 

a  second  which  the  moon's  right  ascension  increases  in  one 
second  of  time  at  the  instant  July  5th,  oh.  In  the  hourly 
ephemeris  of  the  moon  given  in  the  Nautical  Almanac  there 
is  given  in  connection  with  the  moon's  right  ascension  the 
"  difference  for  one  minute,"  which  is  simply  the  value  of 
the  differential  coefficient  multiplied  by  60  ;  i.e.,  we  may  sup- 

pose the  a  in  -7—  to  be  expressed  in  seconds,  and  the  T  in 

minutes.  Thus  we  have  for  the  example  above  the  "  differ- 
ence for  one  minute"  =  2S.3O346.  So  in  connection  with  the 
solar  ephemeris  there  is  given  the  sun's  hourly  motion  in 

right  ascension,  which  is  the  value  of  -^-multiplied  by  60X60. 

The  hourly  motion  in  declination  is  expressed  in  seconds  of 
arc. 

52.  By  means  of  these  differential  coefficients  as  given  in 
the  ephemeris,  the  second  differences  are  taken  into  account 
in  the  interpolation  in  a  very  simple  manner,  for  we  have  to 
second  differences  inclusive 


The  difference  of  these  expressions  is 


and 

f(T  +  **)  =  AT)  +  nW  +  -  .   (102) 


Thus  we  have  only  to  correct  the  value  of  the  first  differen- 
tial coefficient  by  adding  to  it  algebraically  the  product  of 


§  53-  DIFFERENTIAL    COEFFICIENTS.  9  1 

the  difference  of  two  consecutive  values  by  one  half  the  in- 
terval n.  We  then  use  the  corrected  differential  coefficient, 
as  we  should  do  if  the  first  differences  were  constant.  % 

Example  9.  Required  the  sun's  right  ascension  and  decli- 
nation, 1883,  July  4th,  4h,  Bethlehem  mean  time. 

As  the  longitude  of  Bethlehem  from  Washington  is 
—  6m4Os.2,  the  corresponding  Washington  time  is  3h53mi9s.8 
=  July  4th,  3h.8888  =  July  4.162. 

From  the  solar  ephemeris  for  the  meridian  of  Washington 
we  then  find  : 

Date.  &••  Hourly  Motion.  o.  Hourly  Motion. 

July  4.0  6h53ra33s79  ios.3O7  22°  52'  51".!  -  I3".i9 
July  5.0  6h57m4is.o2  10^294  22°  47'  22".7  -  14".  18 
^d^a  11 

~       '  2   =  '°13  X  *'        = 


Corrected  hourly  motion  =  ios.3o6 

10.306  X  3h-889  =  40s.o8 
Required  a  =  6h54mi3s.87. 
d*d     n 

~     '  2  ~  '"  x  **   •  ~     ° 


Corrected  hourly  motion  --  I38.27 

13.27  X  3h-8S9  =  51"  61 

»      Required  #  =  22°  51'  59".5. 

53.  If  values  of  the  differential  coefficients  are  required 
for  values  of  the  argument  between  the  dates  of  the  table, 
we  may  derive  the  necessary  formulae  by  differentiating  the 
function  developed  by  Taylor's  formula  (100),  viz.: 


df(T)  , 

~ 


dT  dT~          df  dT     1.2 


<Tf(T)      d'f(T) 


dT        -    dr  dT 


* 


(103) 


92  INTERPOLA  TION.  §  54. 

Substituting  in  these  the  values  of     }-,  -  .     j^~  •  •  .    (101), 
we'  have  the  values  required.* 


The  Ephemeris. 

54.  In  case  the  American  Ephemeris  and  Nautical  Almanac 
is  used,  most  of  the  quantities  there  tabulated  may  be  taken 
from  the  tables  by  the  method  of  Art.  52,  an  example  of  the 
application  of  which  has  been  given.  The  lunar  distances 
which  are  given  in  that  part  of  the  ephemeris  computed  for 
the  meridian  of  Greenwich  form  an  important  exception. 
These  distances  are  given  for  three-hour  intervals,  together 
with  the  "proportional  logarithm  of  the  difference."  This  pro- 
portional logarithm  is  simply  the  logarithm  of  3h  —  the  inter- 
val of  the  table  —  divided  by  the  difference  between  the  two 
consecutive  distances.  It  is  convenient  to  suppose  the  3h  re- 
duced to  seconds  of  time,  and  the  tabular  distance  expressed 
in  seconds  of  arc.  The  proportional  logarithm  may  then  be 
denned  as  the  number  of  seconds  of  time  required  for  the  distance 
to  change  one  second  of  arc.  Thus  : 

1883,  July  6th,  oh,  distance  between  centres 

of  sun  and  moon  =  24°    2'  55" 

1883,  July  6th,  3h,  distance  between  centres 

of  sun  and  moon  =  25°  32'  44" 
Difference  =     i°  29'  49" 

3h 
Proportional  logarithm  of  difference  —  log  to  2  /  .  ~// 


io8ooa 


*  A  very  full  discussion  of  this  subject,  with  elaborate  tables  for  computing 
the  numerical  coefficients,  may  be  found  in  Vol.  II.  of  Oppolzer's  "  Lehrbuch 
zur  Bahnbestimmung." 


§  54-  THE  EPHEMERIS,  93 

For  simple  interpolation,  disregarding  second  and  higher 
orders  of  differences,  we  proceed  as  follows : 

Let  7"and  T-\-  3h  =  the  two  consecutive  dates  between  which 

the  distance  is  to  be  interpolated  ; 
T  -\-  t  =  the  time  for  which  the  distance  is  required; 
D  and  D/  =  the  distances  at  times  T  and  T-\-  3h  ; 
D'  —  distance  at  time  T  -\-  t.\ 
4  =  D,-D; 
A'  =  D'  -  D. 

Then  all  being  expressed  in  seconds, 

A'  :  A  —  t  :  10800; 

log  A'  =  log  /  -  PLA. (104) 

If  we   subtract   both   members  of  this    equation  from  log 
10800,  we  have 

10800  10800    , 

log  —£-  =  log  — —  +  PLA, 

or  PLA'  =  PLt  +  PLA (104), 

With  formula  (104)  only  the  common  logarithmic  tables 
are  required;  with  (104),  we  use  the  tables  of  proportional 
or  logistic  logarithms  given  in  works  on  navigation.  The  lat- 
ter tables  give  at  once  for  any  angle  t  the  logarithm  of 

3h        3° 

V  or  -or.     Sometimes  the  tables  are  computed  for  the  argu- 

ih 
ment  -— . 

The  following  simple  example  will  illustrate  both  formulae 
(104)  and  (104), : 

Example  10.     Required  the  distance  between  the  centres 


94  INTERPOLATION.  §54. 

of  the  sun  and  moon,  1883,  July  6th,  ih  15™,  Greenwich  mean 
time. 

From  the  ephemeris,  1883,  July  6th,  oh,  D   —  24°    2'  55" 

PL  Difference  =     .3019 
t  —  ih  15™  =  4500s          log  t  —  3.6532 

log  A'  =3o5i3.       Therefore  A'  =         37'  25" 

D'  =.  24°  40'  20" 

For  using  equation  (iO4)t  we  employ  the  tables  of  propor- 
tional logarithms  given  in  Bowditch's  Navigator,  Table 
XXII: 

PL  Difference  ==  .3019 
PL  ih  I5m  =  .3802 
PL  A'  =6821;  A'  =  o°  37'  25". 

As  will  be  seen,  with  the  proportional  logarithms  the 
quantity  A'  is  given  at  once  in  degrees,  minutes,  and  seconds, 
without  the  necessity  of  reducing  t  in  the  first  place  from 
the  sexagesimal  to  the  decimal  notation,  and  in  the  second 
place  reducing  A'  from  the  decimal  to  the  sexagesimal.  At 
the  end  of  the  American  Ephemeris  for  1871  is  given  a  table 
of  "  Logarithms  of  small  Arcs  in  Space  or  Time"  by  using 
which  this  reduction  is  also  avoided. 

The  foregoing  process  disregards  second  and  higher 
orders  of  differences.  In  order  to  take  these  into  account, 
we  have  in  the  general  interpolation  formula  (92) 

nw  =  t,  w  —  3h;      /.  n  —  -T. 


In  which  A"  will  be  the  difference  between  two  consecutive 
values  of  A. 


54-  THE  EPHEMERIS.  95 


---- 

t  t  2h   —   t          \ 

and  formula  (92),  becomes     U  =  D  +  —  ^J  -     ?-g—  A"). 

I          ^  —  t      \ 
Let     IJ  —  —  z  —  ^"  )  =  \A\  —  corrected  tabular  difference  ; 

Q  =  PLA>        [<2]  =  PL\A\. 
Then  we  may  assume 

/          ^  _  t      \ 

\Q  —  ^-g  —  Q")  =  IQ]  with  sufficient  accuracy,'  (105) 


in  which  Q  ''  is  the  difference  between  two  consecutive  val- 
ues of  Q.  (Q  and  A  are  inverse  functions  one  of  the  other, 
but  the  algebraic  sign  of  the  correction  need  give  no 
trouble.) 

It  will  be  a  little  more  accurate  if  we  take  for  Q'  the 
arithmetical  mean  of  the  differences  between  Q  and  both  the 
preceding  and  following  values  found  in  the  table. 

Example  1  1.  Required  the  distance  between  the  centre  of 
the  moon  and  Fomalhaut,  1883,  Jwty  2Otn>  I9h  2Om  5s,  Gh. 
M.  T. 

From  the  ephemeris, 

July  20th,  I5h  G=-4536     £"=  +  211 

July  20th,  i8h      D  32°  41'.  20"      Q  =  .4747 
July  2oth,  2ih      D  31°  41'    o"      Q  =  .4995     &'     '~  +  248 

Then    /     =  ih  2om  5"  =  ih-3347  [Q]  =    .4683  A'  =    o°  27'  14".  5 

Mean  Q"  =       230          log    t   =  3.6817  D'  =  32°  14'    5".  5 


log  A>  =  3-2134 


If  we  had  neglected  the  second  differences  in  this  example 
we  should    have  found  A'  =  o°  26'  51",  which  can  only  be 


g6  INTERPOLATION.  §55. 

considered  a  rough  approximation.  If  the  interpolation  be 
extended  to  third  differences,  we  find  A'  =  27'  13".%.  This 
differs  from  the  first  value  by  a  quantity  which  will  be  of 
very  little  importance  in  practical  cases. 

To  Find  the  Greenwich  Time  Corresponding  to  a  Given  Lunar 

Distance. 

55.  First.  We  may  interpolate  the  time  directly  from  the 
ephemeris,  neglecting  the  second  differences;  then  with  the 
time  so  found  as  a  first  approximation  we  deduce  the  cor- 
rected proportional  logarithm  [<2],  and  repeat  the  computa- 
tion. 

t  being  the  required  quantity,  either  (104)  or  (104),  give 
the  first  approximation,  viz., 

log  /  =  log  A1  +  PLA, (106) 

or  PLt  =    PLA'  -  PLA (106), 

Then  with  this  value  of  t  we  determine  the  corrected  pro- 
portional logarithm  [Q]  by  (105),  and  repeat  the  computation. 

Example  12.  1883,  July  2Oth:  determine  the  Gh.  M.  T. 
when  the  distance  between  the  moon's  centre  and  Fomal- 
haut  was  32°  14'  5".5. 

.4536 

We  find  from  the  ephemeris  that  on  July  2Oth,  i8h  D   =  32°  41'  20"    PL  .4747 

Given  value  of  D'  —  32°  14'    5".  5         .4995 
log^'  =  3.2134  Therefore  A'  —         27'  14". 5 

PLA    =    .4747 
log    /   =  3.6881  Approximate  t   —  ih  2im  i68 

By  (105),  -  ^-=-V=  ~  63-  Therefore  [0]    =    .4684  =  PLA 

Repeating  computation,  PLA    =    .4684 

log  A'  =  3.2134 

t  =  ih  20m  oo8  log  /      =  3.6818 

Required  Gh.  M.  T.,  July  2oth,  igh  2Om  6*. 


§55-  THE  EPHEMERIS.  97 

# 

Table  I  at  the  end  of  the  American  Ephemeris  gives  the 
correction  required  on  account  of  the  second  differences  in 
the  moon's  motion  in  finding  the  Greenwich  time  corre- 
sponding to  a  given  lunar  distance.  It  is  designed  to  obvi- 
ate the  necessity  for  the  second  computation  in  the  case 
just  considered.  The  formula  for  this  correction  is  derived 
as  follows  : 

Let      T  +  t   =  the  time  taken  from  the  table  when  second 

differences  are  neglected  ; 
T  +  t'  =  the  time  taken  when  second  differences  are 

considered  ; 

Q  and  [<2]    —  the  tabular  and  corrected  proportional  log- 
arithms. 

Then  (io6)log  /  =  log  A'  +  Q; 
log  *'=  log  A'  +  [<2]  ; 

log  t'-  log  /  =  [<2]-<2  -  -          -G",  from  (105). 


Then  as  log  t'  —  log  t  will  never  be  very  large,  we   may 
treat  it  as  a  differential,  viz., 


log  t'  -  log  /  =  Jlog  /  = 
M  being  the  modulus  =  .434294. 

Then 


«...    (107) 


Where    t    is    supposed    given    in    minutes    and   /'  —  /   is 
expressed  in  seconds.     The  correction   will  be   applied  to 


$  INTERPOLA  TION.  §  56. 

with  th  e  |  J^us  [  sign  when  the  proportional  logarithm 


.     j  diminishing  ) 
'    (   increasing    j  " 

If  the  table  is  not  at  hand,  t'  —  t  may  very  readily  be 
computed  from  (107). 

In  the  last  example,  /     =  ih  2im  i6s  =  8im.267; 

Q"  =  230. 
Therefore  tf  —  t  =         —  ira  ios.8; 

/  —          Ih  20m     58.2. 

56.  In  the  British  Nautical  Almanac  the  differential  co- 
efficients are  not  given  in  connection  with  the  right  ascen- 
sion and  declination  of  the  sun,  moon,  and  other  bodies  as 
in  the  American  Ephemeris.  If,  therefore,  it  is  considered 
necessary  to  carry  the  interpolation  to  second  differences,  it 
must  be  done  by  the  interpolation  formula. 


PRACTICAL   ASTRONOMY, 


CHAPTER    I. 

THE   CELESTIAL  SPHERE.— TRANSFORMATION  OF 
CO  ORDINATES. 

57.  When  we  view  the  heavens  on  a  clear  night,  the  stars 
and  other  celestial  bodies  appear  to  us  to  be  projected  on  the 
surface  of  a  sphere  of  indefinite  radius,  with  the  centre  at  the 
eye  of  the  observer. 

A  few  hours'  observation  would  show  us  that  all  these 
bodies  are  apparently  revolving  about  us  from  east  to  west, 
in  such  a  manner  as  to  make  a  complete  revolution  in  about 
twenty-four  hours.  This  appearance  we  know  from  other 
considerations  is  due  to  the  diurnal  revolution  of  the  earth. 

In  addition  to  this  first  motion  we  should  soon  recognize 
a  second,  in  consequence  of  which  the  sun  appears  to  move 
among  the  stars  from  west  to  east,  in  such  a  manner  as  to 
complete  a  revolution  in  about  one  year.  We  know  this  to 
be  due  to  the  annual  revolution  of  the  earth  about  the  sun. 
There  are  various  other  motions  recognized,  some  of  which 
require  very  long  periods  for  completing  their  cycle.  Of 


IOO  PRACTICAL  ASTRONOMY.  §  58. 

these  precession  and  nutation  are  examples.  Some  of  these 
motions  we  shall  have  occasion  to  consider  hereafter. 

For  our  purposes  it  will  frequently  be  convenient  to  speak 
of  the  apparent  motions  of  the  heavenly  bodies  as  if  they 
were  the  true  motions.  Thus  we  say  that  a  star  passes  the 
meridian  at  a  given  time,  when  we  know  in  fact  that  the 
meridian  passes  the  star;  or  that  the  sun  rises  above  the 
horizon,  when  in  fact  the  horizon  passes  below  the  sun.  The 
reader  will  never  be  misled  by  such  expressions,  and  we  are 
by  this  means  often  able  to  avoid  cumbersome  circumlocu- 
tions in  language. 

As  we  view  the  celestial  sphere  all  the  heavenly  bodies 
appear  to  be  at  equal  distances,  and  with  few  exceptions  to 
maintain  the  same  positions  relative  to  each  other.  We  can 
measure  their  directions;  but  at  present  are  not  concerned 
with  their  distances. 

The  department  of  astronomy  with  which  we  are  now 
occupied  deals  for  the  most  part  with  exact  measurements — 
either  of  the  co-ordinates  of  the  stars,  or  of  the  observer's 
position  on  the  earth's  surface.  If  we  know  the  latitude  and 
longitude  of  our  observatory,  we  can  by  observation  deter- 
mine the  spherical  co-ordinates  of  any  star.  If,  on  the  other 
hand,  the  positions  of  the  heavenly  bodies  are  known,  obser- 
vation furnishes  the  data  for  determining  our  position  in 
latitude  and  longitude.  It  is  with  problems  of  the  latter 
class  that  this  book  is  chiefly  concerned. 

Spherical  Co-ordinates. 

58.  The  position  of  a  star  on  the  celestial  sphere  is  deter- 
mined by  means  of  two  spherical  co-ordinates,  measured  with 
reference  to  a  fixed  great  circle. 

Three  different  systems  are  in  common  use,  according  as 
the  circle  of  reference  is  the  horizon,  the  equator,  or  the 


§  58.  SPHERICAL    CO-ORDINATES.  IOI 

ecliptic.     For  our  purposes  we  shall  define  these  circles  as 

follows: 

THE  HORIZON  is  a  great  circle  of  the  celestial  sphere  formed 
by  a  plane  passing^  through  the  eye  of  the  observer  and  per- 
pendicular to  the  plumb-line. 

THE  CELESTIAL  EQUATOR  is  a  great  circle  of  the  celestial 
sphere  formed  by  a  plane  passing  through  the  eye  of  the  ob- 
server and  perpendicular  to  the  earths  axis. 

THE  ECLIPTIC  is  a  great  circle  of  the  celestial  sphere  formed  by 
a  plane  passing  through  the  eye  of  the  observer  and  parallel 
to  the  plane  of  the  earths  orbit. 

Either  of  these  circles  considered  as  the  basis  of  a  system 
of  co-ordinates  is  called  a  primitive  circle.  The  great  circles 
formed  by  planes  perpendicular  to  the  primitive  circle  are 
called  secondaries. 

THE  ZENITH  is  the  point  where  the  plumb-line  produced  pierces 

the  celestial  sphere  above  the  horizon. 
THE  NADIR  is  the  point  where  the  plumb-line  produced  below 

the  horizon  pierces  the  celestial  sphere. 
THE  ZENITH  and  NADIR  are  the  poles  of  the  horizon. 

Vertical  circles  are  secondaries  to  the  horizon. 
Hour-circles,  or  circles  of  declination,  are  secondaries  to 
the  equator. 

THE  MERIDIAN  is  the  hour-circle  which  passes  through  the 
zenith  and  nadir. 

THE  MERIDIAN  LINE  is  the  line  in  which  the  plane  of  the 
meridian  intersects  the  plane  of  the  horizon.  The  north  and 
south  points  of  the  horizon  are  the  points  in  which  this  line 
pierces  the  celestial  sphere. 

THE  PRIME  VERTICAL  is  the  great  circle  whose  plane  is  per- 
pendicular to  the  plane  of  the  meridian,  and  passes  through 
the  zenith. 


102  PRACTICAL   ASTRONOMY.  §  59. 

THE  EAST  AND  WEST  LINE  is  the  line  in  which  the  plane  of  the 
prime  vertical  intersects  the  plane  of  the  horizon.  The  east 
and  west  points  of  the  horizon  are  the  points  in  which  this 
line  pierces  the  celestial  sphere. 

The  north  and  south  points  are  the  poles  of  the  prime 
vertical. 

The  east  and  west  points  are  the  poles  of  the  meridian. 

The  Horizon. 

59.  The  spherical  co-ordinates  referred  to  the  horizon  as 
the  primitive  or  fundamental  plane  are  the  altitude  and  azi- 
muth. 

THE  ALTITUDE  of  a  heavenly  body  is  its  distance  above  the 
horizon,  measured  on  a  vertical  circle  passing  through  that 
body. 

THE  AZIMUTH  of  a  heavenly  body  is  the  distance  from  the  north 
or  south  point  of  the  horizon,  measured  on  the  horizon  to  the 
foot  of  the  vertical  circle  passing  through  the  body. 

For  astronomical  purposes  it  is  customary  to  measure  the 
azimuth  from  the  south  point  through  the  entire  circumfer- 
ence in  the  order  S.,  W.,  N.,  E.  For  geodetic  purposes  it  is 
generally  reckoned  from  the  north  point.  Navigators  and 
surveyors  frequently  use  other  methods,  which  it.  is  not 
necessary  to  enlarge  on  in  this  place. 

Instead  of  the  altitude,  the  zenith  distance  of  a  star  is  fre- 
quently used ;  this  is  simply  the  distance  from  the  zenith  to 
the  star,  measured  on  a  great  circle.  The  zenith  distance  and 
altitude  are  complements  of  each  other. 

We  shall  use  the  following  notation  : 

h  =  altitude ; 
a  =  azimuth ; 
z  =  zenith  distance.  z  =  90°  —  h. 


§  60.  THE  EQUATOR.  103 

In  consequence  of  the  diurnal  motion  the  altitude  and  azi- 
muth of  any  star  are  constantly  changing  their  values. 


The  Equator. 

60.  The  points  in  which  the  meridian  intersects  the  equa- 
tor are  the  north  and  south  points  of  the  equator.  The  points 
in  which  the  earth's  axis  pierces  the  celestial  sphere  are  the 
poles  of  the  equator,  and  are  called  respectively  the  north 
and  south  pole.  This  line  is  also  the  axis  of  the  heavens. 

When  the  equator  is  the  fundamental  plane,  the  position 
of  a  star  may  be  fixed  either  by  its  declination  and  hour- 
angle  or  by  its  declination  and  right  ascension. 

THE  DECLINATION  of  a  star  is  its  distance  north  or  south  of 
the  equator  measured-  on  an  hour-circle  passing  through  the 
star.  When  the  star  is  north  of  the  equator  the  declination 
is  -f-;  when  south,  — . 

THE  HOUR- ANGLE  of  a  star  is  the  angle  at  either  pole  between 
the  meridian  and  the  hour-circle  passing  through  the  star  ; 
or  it  is  the  distance  measured  on  the  plane  of  the  equator 
from  the  south  point  of  the  equator  to  the  foot  of  the  hour- 
circle  passing  through  the  star. 

The  hour-angle  is  reckoned  from  the  south,  in  the  direction 
S.,  W.,  N.,  E.,  from  o°  to  360°,  or  from  oh  to  24*.  In  some 
cases  it  is  convenient  to  reckon  the  hour-angle  towards  the 
east,  in  which  case  it  must  be  considered  minus.  The  hour- 
angle  is  constantly  changing,  in  consequence  of  the  appar- 
ent revolution  of  the  celestial  sphere.  As  this  revolution 
does  not  affect  the  position  of  the  equator,  the  declination  is 
independent  of  the  diurnal  motion. 

The  planes  of  the  equator  and  ecliptic  intersect  each  other 


104  PRACTICAL  ASTRONOMY.  §  6l. 

at  an  angle  of  about  23°  27'.  The  line  in  which  these  planes 
intersect  is  the  line  of  the  equinox,  and  the  points  where  it 
pierces  the  celestial  sphere  are  the  equinoctial  points.  They 
are  known  respectively  as  the  vernal  equinox  and  the  autum- 
nal equinox.  The  points  on  the  equator  90°  from  the  equi- 
noctial points  are  the  solstices,  known  as  the  summer  solstice 
and  the  winter  solstice.  The  equinoctial  colure  is  the  hour- 
circle  passing  through  the  equinoxes.  The  solstitial  colure 
is  the  hour-circle  passing  through  the  solstices. 

The  equinoxes  are  the  poles  of  the  solstitial  colure,  and  the 
solstices  are  the  poles  of  the  equinoctial  colure. 

THE  RIGHT  ASCENSION  of  a  star  is  the  nrc  of  the  equator  in- 
tercepted between  the  vernal  equinox  and  the  foot  of  the  hour- 
circle  passing  through  the  star.  It  is  reckoned  from  the  ver- 
nal equinox,  in  the  order  of  the  signs  Aries,  Taurus,  etc., 
from  o°  to  360°,  or  from  oh  to  2$. 

The  right  ascension  and  declination  are  both  independent  of 
the  diurnal  motion.  Instead  of  the  declination,  the  north-polar 
distance  is  frequently  employed.  It  is  the  distance  from  the 
north  pole  to  the  star  measured  on  a  great  circle,  and  is  the 
complement  of  the  declination.  We  shall  let 

8  =  Declination  of  a  star; 
a  =  Right  ascension  ; 
t  =  Hour-angle ; 
/  =  North-polar  distance  =  90°  —  3. 


The  Ecliptic. 

61.  When  the  ecliptic  is  the  fundamental  plane,  the  co- 
ordinates are  called  latitude  and  longitude. 


62. 


THE  ECLIPTIC. 


105 


THE  LATITUDE  of  a  star  is  its  distance  north  or  south  of  the 
ecliptic  measured  on  a  secondary  to  the  ecliptic.  When  north 
of  the  ecliptic  the  latitude  is  +;  when  south,  —  . 

THE  LONGITUDE  of  a  star  is  the  distance  measured  on  the  eclip- 
tic from  the  vernal  equinox  to  the  foot  of  the  secondary  pass- 
ing through  the  star.  It  is  reckoned  in  the  order  of  the  signs 
from  o°  to  360°. 

Longitude  will  be  designated  by  A  ; 
Latitude  will  be  designated  Ipy  /?. 

These  co-ordinates  must  not  be  confounded  with  terrestrial 
latitude  and  longitude,  with  which  they  have  no  connection. 
The  system  is  much  used  in  orbit  computation. 

Fig.  i  will  serve  to  illustrate  the  preceding  definitions. 
It  represents  the  sphere  projected  on  the  plane  of  the  hori- 
zon. 

Zis  the  zenith,  CVTihe  ecliptic,  WVE  the  equator,  O  the 
position  of  any  star. 


OL  =  Declination,  d  ; 
=LPQ=  Hour-angle,  /; 
VEQ  WL  =  Right  ascension, 
VTCD  =  Longitude,  A  ; 
OD  —  Latitude,  ft  ; 
OH  =  Altitude,  h  ; 
SH  =  Azimuth,  a  ; 
OZ  =  Zenith  distance,  z 
PO  —  N.  P.  distance,/. 


FIG.  i. 


62.  The  following  diagram  will  assist  in  giving  definiteness 
to  the  symbols  employed  in  the  foregoing.      The  notation 


io6 


PRACTICAL  ASTRONOMY. 


should  be  thoroughly  memorized,  as  the  symbols  will   be 
constantly  employed  hereafter. 


f  Azimuth  —  a-, 
Horizon^  Altitude  =  h  ; 

[Zenith  distance  =  z. 


SphericalCo-ordinates 


f  Hour-angle  =  /  ; 

Equator  J  Right  ascension  =  «; 

Declination  =  d ; 
[  North-polar  distance  =  /- 


The  obliquity  of  the  ecliptic  we  shall  designate  by  f.  Its 
mean  value  for  1881.0  is  e  =  23°  27'  16". 60.  (See  American 
Ephemeris,  page  248.) 

The  position  of  the  observer  on  the  surface  of  the  earth  is 
given  in  latitude  and  longitude.  We  shall  let 

cp  =  Latitude,  -f-  when  north,  —  when  south; 
L  =  Longitude,  -|-  when  west,  —  when  east. 

63.  For  astronomical  purposes  longitude  in  this  country 
is  reckoned  from  the  meridian  of  Washington  or  Greenwich. 

In  Fig.  2  the  large  circle  represents  a  section  of  the  celes- 
tial sphere,  and  the  small  one  a  section  of  the  earth,  both 
formed  by  the  intersection  of  the  plane  of  the  meridian. 
HH'  is  the  horizon,  RE1  the  equator,  Z  the  zenith,  Z' 
the  nadir,  P  the  north  pole. 

The  latitude  of  the  point  'O  will  be  equal  to  the  arc  EZ, 
which  by  definition  is  the  declination  of  the  zenith  of  O.  It 
is  also  equal  to  the  arc  PHf,  or  the  elevation  of  the  north 
pole  above  the  horizon  of  O. 


§64. 


TRANSFORMATION  OF  CO-ORDINATES. 


107 


The  angle  between  the  equator  and  the  horizon  of  any  place 
will  therefore  be  90°  —  q>,  y>  being  the  latitude  of  the  place. 


Transformation  of  Co-ordinates. 

64.  PROBLEM  I.  Having  given  the  altitude  and  azimuth  of 
any  star,  to  find  the  corresponding  declination  and  hour-angle. 

Let  us  refer  the  star's  position  to  a  system  of  rectangular 
co-ordinates  in  which  the  horizon  shall  be  the  plane  of  XY, 
the  positive  axis  of  X  being  directed  to  the  south  point,  the 
positive  axis  of  Y  to  the  west  point,  and  the  positive  axis  of 
Z  to  the  zenith. 

Then  will  x,y,  z  =  the  rectangular  co-ordinates  of  the  star; 

A,  h<  a  =  the  polar  co-ordinates  of  the  star; 
A  being  the  distance  or  radius  vector. 

We  then  have*       x  =  A  cos  h  cos  a\  \ 

y  —  A  cos  h  sin  a\  (  ..... 

z  =  A  sin  h. 


*  See  Davies'  Analytical  Geometry,   edition  of  1869,   p.   302;  or  any  other 
work  on  analytical  geometry  of  three  dimensions. 


108  PRACTICAL   ASTRONOMY.  §  64. 

Let  the  star  now  be  referred  to  the  equator  as  the  funda- 
mental plane,  the  positive  axis  of  X  being  directed  to  the 
south  point  of  the  equator,  the  positive  axis  of  Fto  the  west 
point,  and  the  positive  axis  of  Z  to  the  north  pole. 

Let  now  x ',  y,  z'  be  the  rectangular  co-ordinates; 
A,  #,  t  be  the  polar  co-ordinates. 

We  then  have  x'  =  A  cos  $  cos  /;  \ 

y  =  A  cos  $  sin  *;   > (ill) 

z'  —  A  sin  8.  ) 

The  problem  now  requires  these  values  of  x' ,  y' ,  and  z'  to 
be  expressed  in  terms  of  x,  y,  and  z.  We  observe  that  the 
axes  of  Fare  the  same  in  both  systems;  that  the  axes  of  X 
and  Z  make  the  angle  90°  -  cp  with  those  of  X'  and  Z'. 
We  therefore  require  the  formulas  for  transformation  of  co- 
ordinates from  one  rectangular  system  to  another  having  the 
same  origin,  viz.: 

x'  =       x  cos  (90°    -  <p) -\-  z  sin  (90°  —  cp); 

y  =       /I    : 

z'  =  —  x  sin  (90°  —  cp)  -f-  z  cos  (90°  —  cp); 

or  x'  =        x  sin  cp  -f-  z  cos  cp\  \ 

y'  =     y\  \ (112) 

z'  =  —  x  cos  cp  -\-  z  sin  cp.  ) 

Substituting  in  (112)  the  values  of  x,  y,  and  z  from  (no), 
and  of  y, y,  and  z'  from  (in),  dropping  at  the  same  time 
the  factor  A  which  is  common  to  every  term,  we  have  • 

cos  d  cos  t  =        cos  h  cos  a  sin  cp  -f-  sin  ^  cos  99;  \ 
cos  #  sin  /  =        cos  h  sin  #;  I  (113) 

sin  #  =  —  cos  h  cos  #  cos  cp  -(-'  sin  //  sin  <p.  ) 


§64.  TRANSFORMATION  OF  CO-ORDINATES.  1 09 

These  equations  express  the  required  relation,  but  they 
are  not  in  convenient  form  for  logarithmic  computation;  be- 
sides, the  required  quantities  8  and  t  are  given  in  terms  of 
their  sines  and  cosines. 

It  is  always  best,  when  practicable,  to  determine  an  angle 
in  terms  of  its  tangent.  The  tangent  varies  rapidly  for  all 
angles  great  or  small,  and  consequently  if  a  small  error  from 
any  cause  exists  in  the  tangent  it  will  have  but  little  effect 
on  the  value  of  the  angle.  -  On  the  other  hand,  if  the  value 
of  the  angle  is  near  90°  or  270°  and  is  given  in  terms  of  its 
sine,  this  function  will  vary  slowly  with  the  angle,  and  a 
small  error  in  the  sine  will  produce  a  large  error  in  the 
angle.  The  same  is  true  of  the  cosine  for  angles  near  o°  or 
1 80°.  If  the  angle  is  near  90°  or  270°  it  may  be  determined 
with  accuracy  from  its  cosine,  or  if  near  o°  or  180°  it  may  be 
accurately  determined  from  its  sine.  In  any  case  it  can  be 
determined  with  accuracy  from  its  tangent. 

For  the  purpose  of  effecting  the  required  transformation  in 
(113),  let  us  introduce  the  auxiliary  equations 

sin  h  —  n  cos  A'; 
cos  h  cos  a  =  n  sin  N. 

This  will  be  possible,  for  we  have  the  two  arbitrary  quan- 
tities n  and  N,  and  the  two  equations  (114)  for  determining 
them.  Substituting  these  values  in  (113),  we  have 

cos  S  cos  t  =       n  sin  N  sin  cp  -\-  n  cos  N  cos  cp  =  n  cos  (q>  —  A7");  ) 

cos  d  sin  /  =  cos  h  sin  a\       >  (115) 

sin  d  =  —  n  sin  YVcos  <p  -f-  n  cos  AT  sin  cp  =  n  sin  (<p  —  TV).  / 

For  determining  N  we  divide  the  second  of  (114)  by  the 
first,  then  we  have 

tan  N  =  cot  h  cos  a.     .     .     .    -     -       116} 


HO  PRACTICAL   ASTRONOMY.  §64. 

For  determining  /  we  divide  the  second  of  (115)  by  the 
first,  and  substitute 

cos  h  cos  a 
sin  N 


from  (114),  viz.,     tan  t  =  CO_N\  tan  a  .....     (117) 

For  determining  tf,  divide  the  third  of  (115)  by  the  first: 
tan  S  —  tan  (<p  —  N)  cos  t  .....     (i  18) 

We  may  now  obtain  a  formula  for  proving  the  accuracy 
of  the  computation  by  dividing  the  second  of  (114)  by  the 
first  of  (115),  viz., 

sin  N  cos  h  cos  a 


cos  ((p  —  N)       cos  #  cos  /' 

Formulae  (116),  (117),  and  (118)  solve  the  problem  com- 
pletely, and  (119)  is  a  proof  of  the  accuracy  of  the  work. 
The  proof  consists  in  this  equation  being  satisfied  when  we 
substitute  for  tf  and  t  the  values  obtained  from  equations  (117) 
and  (118).  If  the  work  has  been  correctly  performed  the 
two  logarithms  should  not  differ  by  more  than  three  or  four 
units  in  the  last  place.  This  proof  is  not  always  reliable, 
however. 

Collecting  together  these  formulae  for  convenience  of 
reference,  we  have 

tan  N  —  cot  h  cos  a\ 

sin  N 

tan  a; 


cos  (<p  -  N} 
tan  d  =  tan  (q>  —  N)  cos  /; 
sin  N    _  cos  h  cos  a 
cos  (cp—N)  ~~  cos  d  cos  f 


§64.  TRANSFORMATION  OF  CO-ORDINATES.  Ill 

With  regard  to  the  species  of  these  angles  it  is  to  be  re- 
marked, first,  Nmay  be  taken  in  any  quadrant  which  satisfies 
the  algebraic  sign  of  tan  TV;  second,  3  is  always  less  than  90° 
and  is  -f-  when  tan  3  is  -)-,  and  —  when  tan  is  — ;  third,  for 
the  species  of  /  let  us  examine  the  equation 

cos  3  sin  t  —  cos  h  sin  a. 

* 

Cos  3  and  cos  h  will  always  be  +,  therefore  the  species  of  t 
will  be  the  same  as  that  of  a. 

As  an  example  of  the  application  of  these  formulas,  take 
the  following: 

Latitude  of  Sayre  Observatory  =  (p  =  40°  36'  25"  .g\ 
Sun's  altitude  =  h  =  47°  15'  i8".3; 
Azimuth  =  a  =  80°  23'    4/r.47; 

Required  #  and  /.     The  computation  is  as  follows : 

<p.  =  40°  36'  23". 9 

h  =  47°  15'  i8".3       cot  h    =  9.9657782  cos  h  =  9.8317007 

a  =  80°  23'    4". 47     cos  a    =  9.2228053  COS  a  =  9.2228053 

N=    8°  46'  33". 2       tan  N=  9.1885835  9.0545060 

<p  —  W=3i°  49'  50". 7 
/  =  46°4o'    4". 53 
S  =  23°    4'  24".33 

tan  a  =  0.7710501 
sin  N  =  9.1834690 
sec  ((p  —  N)  =    .0707805     tan  (q>  —  N]  —  9.7929304 

tan  t  =    .0252996  cos  /  =  9.8364670     cos      =  9.8364670 

tan  $  =  9.6293974     cos  d  =  9.9637894 

9.8002564 

sin  N  _  cos  h  cos  a 

-*-*  =  9-2542495  (proof)  -——-—-  =  9.2542496 


cos  (<p  -  N)  '  cos  d  cos  / 


112 


PRACTICAL   ASTRONOMY. 


65.  PROBLEM  II.  Having  given  the  declination  and  hour- 
angle  of  any  star,  to  determine  the  altitude  and  azimuth.  This 
is  the  converse  of  the  preceding  problem.  In  this,  case  we  require 
the  values  of  x,  y,  z  in  terms  of  the  values  of x' ',  y',  z' . 

Our  formulae  (112)  for  transformation  then  become 

x  =  x'  sin  cp  —  z'  cos  cp ;  \ 

y=y';  I  .    .    .    .    (120) 

z  =  x'  cos  cp  -|-  z'  sin  cp .  ) 

Substituting  in  these  the  values  of  x,  y,  z,  x' ,  y' ,  z' ,  from  (i  10) 
and  (in),  dropping  at  the  same  time  the  common  factor  J, 
we  have 

cos  h  cos  a  =  cos  <5  cos  /sin  cp  —  sin  6  cos  cp ;  \ 
cos  h  sin  #  =  cos  8  sin  /;  I  (121) 

sin  h  =  cos  tf  cos  /  cos  cp  -|-  sin  #  sin  <p .  ) 

We  may  now  adapt  these  equations  to  logarithmic  computa- 
tion by  introducing  the  auxiliaries  m  and  M,  such  that 

sin  d  =  m  sin  J/; 
cos  d  cos  /  =  m  cos  J/ ; 

when,  by  a  process  like  that  used  in  solving  equations  (113), 
we  find  the  following  formulae : 


tan  M  = 

tan  a  = 
tan  h  — 


tan  fl 
cos  / 


cosj/ 


tan  /; 


cos 


tan  (cp  — 
cos  tf  cos  / 

sin  (cp  —  M)  ~~  cos  h  cos  a' 


cos  M 


66. 


TRANSFORMATION  OF  CO-ORDINATES. 


The  remarks  in  reference  to  the  species  of  the  angles  in 
formulae  (I)  will  apply  equally  to  (II). 

The  following  example  will  illustrate  the  application  of 
these  formulae : 


Given 


<p  =  40°  36' 


Required  a  and  h. 


cp  =  40°  36'  23". 9 

&  =  23°    4'  24". 3       tan  d    =  9.6293972  cos  d 

t  =  46°  40'    4".  5       cos  t     =  9.8364670  cos  t 

M  =  31°  49'  50".  7  •     tan  M  =  9.7929302 
cp  —  M  =    8°  46'  33". 2 

a  =  80°  23'    4". 47    • 
h  =  47°  15'  i8".3 

tan  t  =  o  0252995 
cos  M  =  9.9292195 
cosec  (cp  —  M)  =  .8165310  tan  (q>  —  M)  =  9.1885835 

tan  a  =  0.7710500        cos  a  =  9.2228053   cos  a 
tan  h  —  .0342218   cos  h 


9.9637894 
9.8364670 

9.8002564 


cos  M 


sin  (cp  —  M) 


,. 
=  .7457505  (proof) 


cos  d  cos  t 


9.2228053 

9.8317007 

9.0545060 

.7457504 


66.  As  may  readily  be  seen,  the  preceding  formulae  and 
many  more  may  be  derived  by  applying  the  equa- 
tions  of  Spherical  Trigonometry  to  the  triangle 
formed  by  the  zenith,  the  pole,  and  the  star.  Thus 
in  the  figure  the  sides  of  the  triangle  are  90°  —  q>, 
90°  —  d  =  p,  and  90°  —  h  —  z.  The  angles  are  /, 
1 80°  —  #,  and  q,  the  angle  at  the  star,  called  the 
parallactic  angle.  When  any  three  of  these  quan- 
tities are  given,  the  determination  of  any  other 
part  is  merely  a  question  of  trigonometry.  FlG<  3> 


114  PRACTICAL   ASTRONOMY  §66. 

COROLLARY.  To  find  the  hour-angle  oj  a  star  when  in  the 
horizon,  or  at  the  time  of  rising  or  setting. 

When  the  star  is  in  the  horizon  the  altitude,  h,  is  zero,  and 
the  last  of  equations  (121)  becomes 

cos  $  cos  /  cos  cp  ~\-  sin  8  sin  cp  =  o, 

sin  #  sin  cp 
or  cos  /  =  -  CQS  s  CQS  9  =  -  tan  *  tan  9.  .    .    (122) 

From  this  equation  we  may  determine  t  ;  but,  as  before  re- 
marked, it  is  better  to  determine  the  angle  from  its  tangent. 
For  this  purpose  first  add  both  members  of  (122)  to  unity, 
then  subtract  both  members  from  unity,  and  we  have 

cos  8  cos  (    —  sin  8  sin  c 


cos  d  cos  <p.+  sin  d  sin  cp 
~cos  ~~~     ~~ 

cos   c 


cos   cp  — 

2  sm  '  = 


Dividing  the  second  of  these  by  the  first  and  extracting  the 
square  root, 


At  the  time  of  rising  the  lower  sign  will  be  used ;  at  the 
time  of  setting,  the  upper.  This  formula  may  be  used  to 
compute  the  time  of  sunrise  and  sunset  at  any  place  whose 
latitude  is  known.  For  example,  let  it  be  required  to  com- 
pute the  apparent  time  of  sunrise  at  Bethlehem  on  the  morn- 
ing of  July  4th,  1 88 1. 


§67.        ANGULAR  DISTANCE  BETWEEN   TWO   STARS,         115 

From  the  Nautical  Almanac,  page  329,  we  find  for  the 
sun's  decimation          8  =  22°  52/oi//. 

The  latitude  <p  =  40°  36'  2^'.g. 


cp  —  d  =          17°  44'  22".9  COS  =  9.9788425 

cp  -f-  d  =       63°  28'  24".9  cos  =  9.6499288 

tan"  \t  =    .3289137 

\t  —  -      55°  35'  52^.5  tan  \t  =    .1644569* 

t  =  —  iii°  ii'45".o 

/  =  —  7h  24m  47s. 

It  being  sunrise,  t  is  minus.  If  we  subtract  this  quantity 
from  I2h — the  time  when  the  sun  is  on  the  meridian — we 
have  for  the  apparent  time  of  sunrise 

4h  35m  13s- 

This  differs  from  the  ordinary  or  mean  time  by  an  amount 
equal  to  the  equation  of  time,  as  will  be  explained  hereafter. 
(See  Art.  92.) 

67.  PROBLEM  III.  Required  the  distance  between  two  stars 
whose  rig! tt  ascensions  and  declinations  are  known. 

The  two  stars  and  the  pole  will  form  the  vertices  of  a  tri- 
angle of  which  the  sides  will  be  90°  --  tf,  90°  —  tf',  and  d,  the 
required  distance.  The  angle  opposite  d  will  0 
be  a'  —  a. 

a  and  OL'  are  the  right  ascensions  of  the  stars.    90°£ 
d  and  $'  are  the  declinations. 

In  the  triangle  two  sides  and  the  included 
angle  are  given;  the  third  side  is  required. 


PR  A  C  TIC  A  L   AS  TRONOM  Y. 


§67. 


We  can  apply   equations  (121)   to  this  case  by  writing 
(compare  Figs.  3  and  4) 

h  =  90°—  d\ 


t  =  a'  —  a; 
a  =  180°  -  B. 


Thus  we  have 


sin  d  cos  B  =  sin  3  cos  tf'  —  cos  3  sin  3'  cos  (a'  —  a)  ;  j 

sin  d  sin  B  =  cos  #  sin  (<*'  —  a)  ;  L  (124) 

cos  d  =  sin  d  sin  tf'  +  cos  3  cos  tf'  cos  (a'  —  <x).) 

If  the  quantity  d  can  be  determined  with  sufficient  pre- 
cision from  its  cosine,  the  last  of  these  gives  the  required 
solution,  and  we  may  adapt  it  to  logarithmic  computation  as 
follows  : 

Write  sin  d  =  k  sin  K\ 

COS  d  cos  (a'  —  a)  =  k  cos  K. 


Then 


tan  K  = 

cos  d  = 


tan 


COS  (af  —  of) ' 

sin  d  cos  ($'  —  K} 
sin  K 


.    .    (IV) 


If  this  does  not  give  d  with  the  required  degree  of  accura- 
cy, we  may  determine  it  in  terms  of  the  tangent  in  a  manner 
precisely  similar  to  that  employed  in  solving  equations  (113) 
and  (121).  Thus,  let 

sin  d  =  n  cos  N\ 
cos  d  cos  (a'  —  a)  =  n  sin  N. 


§  6/.         ANGULAR   DISTANCE  BETWEEN   TWO   STARS. 

When  we  readily  find 

tan  N  —  cot  3  cos  (af  —  a); 
tan^  == 


llj 


cos 


sin  JV     * 

TV  tan  («'  —  «); 
' 


sin  N 


cos  (^V  +  <?') 


cos  /> 

cos  £  cos  (af 
sin  d  cos 

Example. 


Required  the  distance  between  the  sun  and  moon,  1881, 
July  4th,  oh,  Bethlehem  mean  time. 

From  the  Nautical  Almanac  for  1881,  p.  114,  we  find,  for 
the  moon, 

a!  —        i2h  39m    3S.22; 
V  =  -    9°  23'  I6"./. 

From  p.  329  of  the  same,  for  the  sun, 


d  =  22°  50'  2 1  ".9. 
The  computation  then  is  as  follows,  using  equations  (IV): 

a'  —  a  =         5h43ra3o'.49 

a'  —  a  =       85°  52'  37". 35  cos  (a1  —  a)  =  8.8567115 

d  =        22°  507  2i".9  tan  d  =  9.6244585  sin  8  =  9.5889992 

K '=       80°  18'  45".  19 
5'  =  -    9°  23'  i6".7 
8'  —  K  -  -  89°  42'     i".89 

d-       89°  52'  55". 5  cos  </=  7.3134726 


=    .7677470         cosecA'=    .0062374 
cos  (3'—  K}  =  7.7182360 


PRACTICAL  ASTRONOMY. 


§67. 


Applying  formulae  (IV),  to  the  solution  of  the  same  prob- 
lem, we  have  the  following: 


cos  =  8.8567115 
cos  =  9.9645407 

8.8212522 


a' 

—  a  = 

85°  52'  37".  35 

cos  =  8.8567115 

d  = 

22°  50'  2l".9 

cot  =  .3755415 

N  = 

9°  41'  i4"-8 

tan  =  9.2322530 

d'  = 

-  9°  23'  i6".7 

N 

-|-  d'  = 

o°  17'  58".! 

B  = 

66°  48'  40".  8 

d  = 

89°  52'  55"-5 

tan  (a'  —  a)  =  1.1421632 
sin  N  =  9.2260154 
cos  (N-\-  d')  =  9.9999940  cot 

factor  =  9.2260214 
tan  B  =  0.3681846 


=  2.2817621 

cos  B  =  9.5952317  cos  =  9.5952317 
tan  d  =  2.6865304  sin  =  9.9999991 


sin  N 


9  5952308 
proof  9.2260214 

=  9'2260214 


CHAPTER  II. 

PARALLAX.— REFRACTION.— DIP  OF  THE  HORIZON. 

68.  The  same  star  may  be  observed  from  points  on  the 
surface  of  the  earth  separated  from  each  other  by  several 
thousand  miles.  If  the  distance  to  the  star  is  so  great  that 
the  diameter  of  the  earth  is  inappreciable  in  comparison,  it 
will  appear  in  the  same  part  of  the  heavens  from  whatever 
part  of  the  earth  it  is  seen.  If,  however,  the  diameter  of  the 
earth  bears  an  appreciable  ratio  to  the  distance  of  the  object, 
then  when  the  observer's  position  changes  there  will  be  an 
apparent  change  in  the  place  of  the  star.  This  difference  in 
position  is  called  parallax. 

It  is  customary  in  dealing  with  bodies  which  have  an  ap- 
preciable parallax  to  reduce  all  positions  to  the  earth's  cen- 
tre. Thus  the  places  of  the  sun,  moon,  and  planets,  which 
we  find  given  in  the  ephemeris,  are  the  places  as  they  would 
appear  to  an  observer  at  the  centre  of  the  earth.  This  which 
we  are  considering  is  the  diurnal  parallax.  With  the  subject 
of  annual  parallax,  which  depends  upon  the  position  of  the 
earth  in  its  orbit,  we  have  at  present  nothing  to  do.  It  may 
be  remarked  that  on  account  of  the  great  distances  of  the 
fixed  stars  their  diurnal  parallax  is  in  all  cases  inappreciable. 
It  is  only  necessary  to  consider  it  in  connection  with  the 
bodies  of  the  solar  system. 


I2O  PRACTICAL  ASTRONOMY.  §  7O. 

Definitions. 
69.  THE  GEOCENTRIC  POSITION  of  a  body  is  its  position  as 

seen  from  the  earth's  centre. 
THE  APPARENT*  or  OBSERVED  POSITION  is  its  place  as  seen 

from  a  point  on  the  earth's  surface. 
THE  PARALLAX  is  the  difference  between  the  geocentric  and  the 

observed  place. 

It  may  also  be  defined  as  the  angle  at  the  body  formed  by 
two  lines  drawn  to  the  centre  of  the  earth  and  the  place  of 
observation  respectively. 
THE  HORIZONTAL  PARALLAX  is  the  parallax  when  the  star  is 

seen  in  the  horizon. 
THE  EQUATORIAL  HORIZONTAL   PARALLAX  is  the  parallax 

when  seen  in  the  horizon  from  a  point  on  the  eartlis  equator. 
It  may  also  be  defined  as  the  angle  at  the  body  subtended 
by  the  equatorial  radius  of  the  earth. 

70.  PROBLEM  I.     To  find  the  equatorial  horizontal  parallax 
of  a  star  at  a  given  distance  from  the  earths  centre. 

Let   n  =  the  equatorial  horizontal  parallax  =  PSC; 
a  =  the  equatorial  radius  of  the  earth  =  PC', 
A  =  star's  distance  from  the  earth's  centre  =  SC. 

\a   Then  from  the  figure  we  have 
IG 

FIG.  5. 

sin7T=~; (125) 

*  The  terms  apparent  place  and  true  place  are  to  be  considered  simply  as 
relative  terms.  When  dealing  with  parallax  we  speak  of  the  true  place  as  the 
place  when  corrected  for  parallax.  So  when  speaking  of  refraction  the  appar- 
ent place  is  the  place  affected  by  refraction,  and  the  true  place  is  the  place  cor- 
rected for  refraction,  but  it  may  still  require  corrections  for  parallax  and  a  va- 
riety of  other  things.  When  dealing  with  the  places  of  the  fixed  stars  we  use 
the  term  apparent  place  in  a  still  different  sense,  as  we  shall  see  hereafter. 


§71- 


PARALLAX. 


121 


s  being  the  place  of  the  star,  /  a  point  on  the  surface  of  the 
earth,  and  c  being  the  centre. 

For  astronomical  purposes  the  mean  distance  of  the  earth 
from  the  sun  is  regarded  as  the  unit  of  measure.  Then  for 
the  sun  we  have 


A  =  i; 


sin  n  =  a . 


(126) 


7i.  PROBLEM  II.  To  find  the  parallax  of  a  star  at  any 
zenith  distance,  the  earth  being  regarded  as  a  sphere. 

In  the  figure,  s  represents  the  place  of  the  star,  z  the  zenith, 
E  the  centre  of  the  earth,  p  a  point  on  the  surface. 

Let 

z'  —  the    observed    zenith 

distance; 

z   —  geocentric  zenith  dis- 
tance; 

p   —  parallax  =  PSE\ 
a   •=.  radius  of  earth  =  PE\ 
A    =  distance  of  star  =  SE. 

From    the   triangle   SEP 
we  have 

A  :  a  —  sin  z'  :  sin  /. 

FIG.  6. 


From  which  sin  p  =  ^  sin  z'\ 

or,  from  (125),  sin  /  =  sin  n  sin  z' (128) 

/  and  it  will  generally  be  very  small ;  hence  for  most  pur- 
poses we  may  write 


=  n  sn  z 


122  PRACTICAL  ASTRONOMY.  §  72. 

The  foregoing  solution  is  only  an  approximation,  the  earth 
not  being  a  sphere  as  we  have  there  regarded  it.  For  many 
purposes  this  is  sufficiently  exact,  while  for  others,  particu- 
larly where  the  moon  is  considered,  it  is  not  so.  A  more 
rigorous  solution  requires  us  to  consider  the  true  form  of  the 
earth. 

Form  and  Dimensions  of  the  Earth. 

72.  The  earth  is  in  form  approximately  an  ellipsoid  of  rev- 
olution, the  deviations  from  the  exact  geometrical  figure  be- 
ing so  small  as  to  be  inappreciable  for  our  purposes. 

The  dimensions  of  the  ellipsoid  as  given  by  Bessel  are  as 
follows: 

Equatorial  radius  A  —  3962.8025  miles; 
Polar  radius  B  =  3949.5557  miles; 
Eccentricity  of  meridian  e  =          .08169683; 

=        8.9122052. 


Many  other  determinations  of  these  quantities  have  been 
made,  differing  more  or  less  from  the  a*bove,  but  these  are 
still  in  more  general  use  than  any  others. 

Definitions. 

73.  THE  GEOGRAPHICAL  LATITUDE  of  a  point  on  the  earth's 

surface  is  the  angle  made  wit/t  the  plane  of  the  equator  by  a 

normal  to  the  surface  at  this  point. 
THE  GEOCENTRIC  LATITUDE  is  the  angle  formed  with  the 

plane  of  the  equator  by  a  line  joining  t/ie  point  with  the 

earth's  centre. 
THE  ASTRONOMICAL  LATITUDE  is  the  angle  formed  with  the 

plane  of  the  equator  by  a  plumb-line  at  the  given  point. 


§73- 


THE  REDUCTION  OF   THE  LATITUDE. 


I23 


If  the  earth  were  a  true  ellipsoid  and  perfectly  homoge- 
neous, the  geographical  and  astronomical  latitude  would 
always  be  the  same.  Practically,  however,  the  plumb-line 
frequently  deviates  from  the  normal  by  very  appreciable 
amounts.  This  deviation  is  always  small,  but  in  mountainous 
countries,  as  the  Alps  and  Caucasus,  deviations  have  been 
observed  as  great  as  29".  Unless  otherwise  stated,  when 
speaking  of  latitude  the  astronomical  latitude  is  to  be  under- 
stood.  We  shall  also  assume  for  present  purposes  that 
it  coincides  in  value  with  the  geographical  latitude. 

Let  the  annexed  figure 
represent  a  section  cut 
from  the  earth's  surface 
by  a  plane  passing  through 
its  axis.  This  section  will 
be  an  ellipse.  Let  K  be 
any  point  on  the  surface,  E) 
P  and  P  the  north  and 
south  poles  respectively. 
Then  HH'  will  represent 
the  horizon  of  the  point  K. 

Let  p   =  CK       =  radius  of  the  earth  for  latitude  KO'E'\ 
<p   =  KO'E'  =  geographical  latitude  of  point  K\ 
<p'  =  KCE'    =  geocentric  latitude  of  point  K\ 
A    =  semi-major  axis  of  ellipse  =  CE'\ 
B   —  semi-conjugate  axis  of  ellipse  =  CP. 


The  angle  CKO  =  q>  —  q>f  is  called  the  reduction  of  the 
latitude.  For  determining  the  parallax  with  precision  we 
require  (<p  —  cpf)  and  p,  the  determination  of  which  for  any 
latitude  <p  we  shall  now  investigate. 


124  PRACTICAL  ASTRONOMY.  §  74. 

To  Determine  (cp  —  cp'). 
74.  We  have  for  the  equation  of  the  ellipse  (Fig.  7) 

A*&-,    .....    (130) 

;    .......     (131) 


<p  being  the  angle  which  the  normal  forms  with  the  trans- 
verse axis  of  the  ellipse.     Also, 

tan?/  =  £  ........    (132) 

By  differentiating  (130)  we  find 
dx 


Therefore  from  (132)  and  (133) 

7?2 
tan  <p'  =  -£  tan  cp  ........    (134) 

From  equation  (134)  cp1  may  be  readily  computed  for  any 
given  value  of  cp.  It  will  greatly  facilitate  this  computation, 
however,  to  develop  (cp  —  cp']  in  the  form  of  a  series.  For 
tfiis  purpose  we  make  use  of  Moivre's  formulae,  viz.:* 

*  As  some  readers  may  not  be  familiar  with  these  very  useful  formulae,  we 
give  their  derivation. 

Developing  n  =  e*  by  Maclaurin's  formula,  we  have 

x         x1  jf3  x* 

**=!+-  +  —  -f—  -H  --  -  -  ,  etc.;.     .     .     (a) 

~    I~1.2    '     I.2.3~I.2.3.4 


also,  cos*  =  I~ 


=.r  ----  1  ----  ,  etc  .......     (c) 

1.2.3       1.2.3.4-5 


§  74.  THE  REDUCTION  OF   THE  LATITUDE.  12$ 


2  COS  X  — 

2  V^-  I  sin  x  = 

V—  i  tan  x  = 


--  e 
—  e 


(135) 


Writing  tan  <pf  =  /  tan  <p  where   /  =  -7,,  substituting  for 

tan   q>'   and  tan  <p  the   value   given   by   the  last  of  (135), 
and   dropping  the  common  factor  V—  i,  we  have 


from  which 


Substituting  in  (^)and  (<r)^s  =  —  z*,     whence     x  —  z  V—  i,     z  =  —  x  V 


we  have 


, 

cos  x  —  i  -] f- 


1.2         1.2.3.4 


./ .  .         «3 

—  r  —  i  sin  #  =  z  A \- 

1.2.3       1.2.3.4.5 


adding,  cos  x—  V^~i  sin  jr  =  i  A 1 1 r  -I {- 

'    I  ~  1.2       1.2.3       1.2.3-4 


etc. 


Writing  —  x  for  -f-  •#>  we  have    cos  x  —  V—  i  sin  x  =  e~x^  — J ; 

cos  x  -\-  V—  i  sin  x  =  *  ^  —1 ; 


adding  and  subtracting, 


2  cos*  =  ^r^-1  4  f~xV~ 
~i  sin*  =  ^^V^T_  g-xV- 


Q.E.D. 


126  PRACTICAL   ASTRONOMY.  §  74- 

Writing  q  —  -    ,      ,  this  becomes 


whence      , » «^(*' -  *)  = -I ^7=-'. (136) 


Taking  the  logarithms  of  both  members  of  equation  (136), 
we  have 

2  V  —  i(<pf  —  cp)  =  log(l  —  ge-2*v-  J)  —  log(i  —  qe^^-~*  ). 

Expanding  the  logarithms  in  the  second  member  by  the 
formula 

X*  X*  X* 

log  (i  -  x)  =     -  *  -  -  -    -  -  -  -,  etc., 

we  have 


6*,   etc. 

• 

This  becomes  by  the  second  of  (135) 

2  V^i  (q>f—  <p)  =  2  V—  i  ^  sin  2cp  +  2  i/—  i  .  |^2  sin  49? 

+  2  I/—  i  fe3  sin  6cp,  etc., 

or  cpf  —  cp  =  q  sin2^?4-fe2  si"  4^+  fe3  sin  6^,  etc.     (137) 


^  _  !       &  -  A* 
In  .this  equation      ^  ==  T~TTf  '  z  r  ^   i    ^- 

Substituting  for  A  and  B  their  values  given  in  Art.  72, 


§  75-  THE  EARTH'S  RADIUS.  I2/ 

and  dividing  by  sin  \"  in  order  to  express  the  result  in  seconds 
of  arc,  we  readily  find 

q   =  -  690^.65; 
tf  -  +       I".i6; 

te3  -  -      "-003. 

Therefore  we  have  the  very  convenient  and  practically  rigor- 
ous  formula 


cp  —  cp'  =  69o".65  sin  2cp  —  i".i6  sin  4<p.     .    (138) 

To  Determine  p. 

75.  x  and  /  being  the  co-ordinates  of  the  point  K,  we  have 

.    (139) 


^2^2;  .......     (130) 

y       B? 
tan  <?/  =  i      ==     -2  tan  cp  ......     (134) 


Combining  (130)  and  (134),  eliminating^,  we  have 


or  ^1  +  tan  9 

Combining  this  with  (139)  and  (134)  to  eliminate  x,  we  find 


sec  qj  /  cos  cp 

P  =  A  —^  =  A  \     -      —r-   —r- -T.  (140) 

Vi  +  tan  cp  tan  cpf  y   cos  ^  cos  (P        W 

The  computation  of  p  from  (140)  is  very  simple,  but  it 
may  be  rendered  much  more  so  by  developing  p,  or   log  p 


128  PRACTICAL  ASTRONOMY.  §  ;6. 

into  a  series.      For  this  purpose  we  shall  regard  A  —  the 
equatorial  radius  —  as  unity,  when  we  have 

see5  <'  l  +  A* 


tan  cp  tan  cp'  &   . 

i  +     -a  tan2  <p       cos2  9  +       sm> 


D4 

Let  us  write       -=«  =  I  — 


Then  we  have 


Taking  the  logarithms  of  both  members, 

2  log  p  =  log  (i  —  g*  sin29>)  —  log  (i  —  e*  sinV). 
Developing  the  second  member  by  the  logarithmic  formula, 

2  loo-  0=-         ~     *  sin      ~  *4  sin^  -          sin      ~  etc' 


or    ogp=         e  —  ^     sn^)  e   -  g    sn 

-  g<)  sin>,  etc. 


Substituting  for  e,  g,  and  M  their  values,  —  J/being  the  mod- 
ulus of  the  common  system  of  logarithms  =  .43429448,  — 
we  readily  find 

log  p  =  —  .00143968  sinV  —  .00001438  sin*(p  —  .00000015  sin6(p.  (141) 

76.  From  this  the  computation  of  log  p  is  very  simple.  A 
better  series  is,  however,  obtained  by  expressing  it  in  terms 
of  functions  of  the  multiple  angles,  instead  of  powers  of  the 
sine  as  here. 


76.  THE  EARTH'S  RADIUS. 

For  effecting  the  required  transformation,  let  us  write  (142) 
log  p  =  a  sin>  +  ft  sin>  +  y  sin>; 


also  sin  <p  =  —7= 

2  r  —  I 


and  for  convenience  write  e*v~  x  =  x\        e-<t><^  =  - 


Then  «  sin 


in>  =  —      ^[f"  2  +  ]?J; 


-  20 


Therefore  log  p  =        3  +  1^+-^+  etc']; 


-[  . 


_L  =  ^2<         __  e-  2          _  2  cos 


~  ^         =  2  cos 


-     =  e^v^~l  -\-  e~  w*^  =  2  cos  6cp. 


130  PRACTICAL  ASTRONOMY.  §78. 

Substituting  these  values  with  the  numerical  values  of  «, 

/?,  and  y  as  given  in  (141),  and  we  find 

\ 

log  p  =  9.9992747  +  .0007271  cos  2cp  —  .0000018  cos4?>.  (142) 
77.  We  therefore  have  for  computing  (cp  —  cpf)  and  log  p, 

<p  —  <p>  =  [2.839258]  sin  2<p  -{-  [o.o6446n]  sin  w,  )        ^ 
log  p  =  9.999  2747  +  [6.861594]  cos  2g>  +  [4-25527*1]  cos  4<p.  )  ' 

In  which  the  quantities  in  brackets  are  logarithms  of  the  co- 
efficients. 

Let  us  apply  formulae  (V)  to  the  determination  of  cp  —  cp' 
and  log  p  for  latitude  40°  36'  23/x.9. 

We  have  2cp  =    81°  12'  48"; 

4cp  =  162°  25'  36". 

9.9992747 

[2.839258]  sin  2<p  =  +  682". 54  [6.861594]  cos  29)  =  ino.6 

[o.o6446n]  sin  4<p  =  —        ".35  [4.25527,1]  cos  4<p  =  17.2 


Therefore        g>  —  q>'  =  n'  22".  19  log  p  —  9.9993875 

78.  We  are  now  prepared  for  the  complete  solution  of  the 
problem  of  parallax.  The  following  method  is  that  of  Olbers 
(see  Bode's  Jahrbuch,  1811,  p.  95). 

We  shall  consider  four  cases,  viz.: 

First — To    determine    the  parallax    in   zenith    distance    and 

azimuth,  having  given  the  geocentric  zenith  distance  and 

azimuth. 
Second — Parallax  in  zenith  distance  and  azimuth,  having  given 

the  observed  zenith  distance  and  azimuth. 
Third — Parallax   in   declination  and  right  ascension,   having 

given  the  geocentric  declination  and  right  ascension. 
Fourth — Parallax    in  declination  and  right  ascension^  having1 

given  the  observed  declination  and  right  ascension.  ' 


§  79-     PARALLAX  IN  AZIMUTH  AND  ZENITH  DISTANCE.     1 3  i 


Case  First. 

79.  Let  the  star  be  referred  to  a  system  of  rectangular 
axes,  the  horizon  of  the  observer  being  the  plane  of  XY,  the 
positive  axis  of  X  being  directed  to  the  south  point,  the 
positive  axis  of  Y  to  the  west  point,  and  the  positive  axis  of 
Z  to  the  zenith. 

Let          £',  ?/,  %  =  the  rectangular  co-ordinates; 
Aft  z ',  a'  =  the  polar  co-ordinates. 

Then  &  =  A'  sin  z'  cos  a'\  \ 

rf  —  A'  sin  z'  sin  a'\  >• (143) 

%  =  A'  cos  *'.  ) 

Next  let  the  star  be  referred  to  a  system  of  co-ordinate 
axes  parallel  to  the  first,  the  origin  being  at  the  centre  of 
the  earth. 

Let  £,  ?;,  £  =  the  rectangular  co-ordinates; 

A,a,z  =  the  polar  co-ordinates; 

and  we  have  g  =  A  sin  z  cos  a\  j 

rf  =  A  sin  z  sin  a\  > (r44) 

%  =  A  cos  z.  } 

Let  the  co-ordinates  of  the  first  origin  referred  to  the 
second  be 

^o»  ^o  ^o  —  rectangular  co-ordinates; 
p,  (cp  —  cp'\  a0  —  polar  co-ordinates. 

With  the  co-ordinate  planes  situated  as  in  the  present  case, 
#0  will  be  zero.  We  shall  write  a0  =  a  —  a,  as  this  form  will 
be  found  convenient  in  a  future  transformation. 


I$2  PRACTICAL   ASTRONOMY.  §79- 

We  then  have 

£o  —  p  sin  (cp  —  cp'}  cos  (a  —  a)-,  \ 

rj.  —  p  sin  (tp  —  (pf)  sin  (a  —  a);  V  .     .     .     (145) 

<20  =  p  cos  (91  —  ?/).  ) 

The  formulas  for  passing  from  the  first  system  (143)  to  the 
second  (144)  will  be 

£'  =  g  -  £0;         T/  =  n  -  T,O;         $'  =  $-  2..     (146) 

Substituting  for  these  quantities  their  values  (143),  (144), 
and  (145),  we  have 

A'  sin  z'  cos  a'=A  sin  #  cos  rt—  p  sin  (cp—cp')  cos  (0—0);  ) 

J'  sin  z'  sin   #'=  J  sin  s  sin  a—  p  sin  (cp—tp')  sin  («—«);  >  (147) 

—p  COS((p—(p').  1 


These  equations  express  the  required  relation  between  the 
quantities  given,  viz.,  a  and  z,  and  those  required,  a'  and  z'  . 
It  remains  to  transform  them  so  as  to  render  their  application 
convenient. 

Let  us  divide  the  equations  through  by  A  and  write  from 


sin  n  =  -,  (a  being  unity  in  this  case.) 

also          f*  =  —r;  viz.: 

/sin  z'  cos  a'  =  sin  z  cos  a  —  p  sin  TT  sin  (cp  —  cp')  cos  (a  —  a);\ 
/sin  z1  sin  a1  =  sin  2  sin  a  —  p  sin  7t  sin  (cp  —  <p')  sin  (a  —  a);  >•  (148) 
/cos  z'  =  cos  z  —  p  sin  ?r  cos  (cp  —  cp1). 

In  these  equations  let  all  horizontal  angles  be  diminished 

*  As/"  is  eliminated  from  our  formulae,  we  are  not  concerned  with  its  value. 


§  79-     PARALLAX  IN  AZIMUTH  AND  ZENITH  DISTANCE.     I  33 

by  a\  the  resulting  equations  will  be  what  we  should  have 
obtained  if  our  original  axes  of  £,  £',  and  £0  had  been  directed 
to  a  point  whose  azimuth  was  a,  instead  of  zero  as  in  the 
present  case.  We  thus  obtain 

/sin  z'  cos  (a!  —  a)  =  sin  z  —  p  sin  n  sin  (cp  —  cp')  cos<z;  )  /  x 
/sin  z'  sin  (a '  —  a)  =  p  sin  n  sin  (cp  —  cp')  sin  a.  \ 

p  sin  n  sin  (cp  —  /n'\ 
Let  us  write      w  = 


sm  # 
Then  (149)  become 

/sin  ^  cos  («'  —  a)  =  sin  #  (i  —  m  cos 
/  sin  zf  sin  (# r  —  a)  =  m  sin  ^  sin  #; 


and  by  division, 


•  ,  ,  m  sin  a 

tan  (a   —  a)  =  — 
i  — 


m  cos  a 


(150)  and  (151)  determine  the  parallax  in  azimuth. 
To  determine  (z'  —  z)  we  proceed  as  follows  : 
Multiply  the  first  of  (149)  by  cos  \(a!  —  a),  and  the  second 

by  sin  \(a'  —  a)  ;  add,  and  divide  the  result  by  cos  %(a  '—  a). 

A  simple  reduction  then  gives 


/sin  z'  =  sin  z  -  p  sin  n  sin  (<p  -  <p')  r  +         (152) 

Let  us  write 


or 


sin  (9>  -  ^     w)  =  cos  (9>  ~  9>}  tan  n 

,  ,,  cos  \(a!  +  ^) 

tan  y  =  tan  (o>  —  cp'}  -     -^—f—  —  (.     .     .     (153) 
;  cos  \(a'  —  a) 


134  PRACTICAL   ASTRONOMY. 

(152)  then  becomes 

y 

/  sin  z'  —  sin  z  —  p  sin  /r  cos  (cp  —  cpf)  tan  7 ; " 
and  the  last  of  (148), 

/  cos  z'  =  cos  z  —  p  sin  n  cos  (cp  —  <p'). 


§  80. 


(154) 


Multiplying  the  first  of  (154)  by  cos  z  and  the  second  by  sin  z, 
and  subtracting,  then  multiplying  the  first  by  sin  z  and  the 
second  by  cos  zt  and  adding,  we  find 


/sin  0'  -  *)  =  p  sin  n  cos  (cp  -  cp') 


/cos  (y—  *)  =  I  -psi 


. 


r 


sn  n  cos    c    — 


TTT  .  . 

Writing  now        »  =  p 

and  we  have 


/sin  (X  —  z)  =  n  sin  (s  —  x)  » 
/cos  (X  —  -8")  =  i  —  n  cos  (^  —  y)  ; 

w  sin  (z  —  y] 
—  z)=.  -        —T  —  ^—- 
i  —  n  cos  (z  —  y) 


. 
(155) 


,     ... 
(156) 


(155)  and  (156)  now  determine  the  parallax  in  zenith  distance, 
and  the  problem  is  completely  solved. 

80.  Formulae  (150),  (151),  (155),  and  (156)  may  be  placed  in 
a  form  more  convenient  for  logarithmic  computation,  as  fol- 
lows :  Write 


p  sin  n  sin  (cp  —  cp'\  cos  a      f      . 
=  mcosa  =  -  -^-  —  .     (157) 

sin  z 


§  80.     PARALLAX  IN  AZIMUTH  AND  ZENITH  DISTANCE.      \  3 $ 

Then 

.  ,  sin  S-  tan  a 

tan  (ft  -  a)  =  -j—^ 

sin  3- 
=  tan 


cos2  %$  —  2  sin  £3"  cos  £3  +  sin2 
sin  5 


=  tan  a 


sin  3  cos  iS  +  sin 

=  tan 


cos2  £3-  -  sin2  -p  '  cosj^  -  sin 
i  +  tan  £3- 


'i-tanty 

But  '  +  tan-f  * 


therefore  , 

tan  (^7  —  a)  =  tan  ^  tan  5  tan  (45°  +  JS).    .     (158) 

In  a  similar  manner  writing 

p  sin  TT  cos  (a)  —  q>f)  cos  (#  —  v)  , 
sin  &  =  n  cos  (^  -  r)  =  -  cosy      "^  -    '  (l59) 

we  find 

sin  5r  tan  (,sr  —  v) 
tan  (zl  —  2)  =  --  -.     ~,   rj 
i  —  sm  3r 

=  tan  3'  tan  (45°  +  JS')  tan  (^  -  7).    (160) 
For  computing  y  we  have 

..cos  \(a'  -\-  a)  /Ncos 

tan  v=  7 


cos  \(a'  —  a)  ^  )     cos  £^/  _ 


136  PRACTICAL  ASTRONOMY.  §  80. 

Therefore 

tan  y  =  tan  (cp  —  cpf)  [cos  a  —  sin  a  tan  \(a!  —  a)~\. 
By  Maclaurin's  formula  we  have 

tan  x  =  x  +  l-ar3,  etc. 

Therefore  if  we  neglect  terms  of  the  third  and  higher  orders 
in  y,  (cp  —  €p'\  and  (of  —  a\  all  of  which  are  small  quantities, 
we  have 

y  —  (q>  —  cpf)  [cos  a  —  sin  a  \(a!  —  a)].      .     (161) 

m  sin  a 

From  tan  (a  —  a]  = 

i  —  m  cos  a 

we  have,  by  neglecting  terms  of  the  higher  orders, 

p  sin  7t(cp  —  cp')  sin  a 

(a1  —  a]  =  m  sin  a  —  -    -—. —    — . 

sin  z 

Substituting  this  in  (161),  we  have 

p  sin  n  sin2#O  —  <p')2  sin  i"     ,  , 

y  —  (cp  —  <p'\  cos  a  — —. — — .    (162) 

2  sin  z 

This  is  accurate  to  terms  of  the  second  !order  of  (cp  —  (p'}  in- 
clusive. 

It  will  readily  appear  that  for  any  value  of  z  not  less  than 
(<p  —  (p'}  the  second  term  will  always  be  inappreciable. 
When  z  is  very  near  zero  the  formula  is  apparently  inappli- 
cable. As  we  shall  not  have  occasion  to  apply  it  to  such 
cases,  it  will  not  be  necessary  for  our  purposes  to  discuss  it 
further.  We  may  therefore  compute  y  from  the  practically 
rigorous  formula 

Y  =  (9  —  <?')  cos  a (163) 


§  8 1.     PARALLAX  IN  AZIMUTH  AND  ZENITH  DISTANCE.      137 

81.  We  have  therefore  the  following  complete  formulae 
for  computing  the  parallax  in  zenith  distance  and  azimuth, 
having  given  the  geocentric  zenith  distance  and  azimuth. 


_  p  sin  n  cos  a  sin  (tp  —  (pr) 

Sill   ~J    '  : • 

sin  z 


tan  (a!  —  a)  =  tan  a  tan  3-  tan  (45°  + 
y  =  (<p  —  (p'}  cos  a; 

p  sin  n  cos  (z  —  y)  cos  (cp  —  cp'} 

sin  -7    = ; 

cos  y 

tan  (zr  —  z]  =  tan  (g  —  y)  tan  5r  tan  (45°  +  %$').  - 


-(VI) 


In  the  meridian,  a  =  a'  —  o.     Therefore  for  this  case  (VI) 
become 

r=<p  —  tp'>t  \ 

sin  y  =  p  sin  TT  cos  [*  -  (<p  -  9')]  (VI), 

tan  (^  -  *)  =  tan  [>  -  (9  -  9')]  tan  S-7  tan  (45 


As  an  example  of  the  application  of  (VI)  let  us  take  the 
•following: 

1881,  July  4th,  Qh,  mean  Bethlehem  time,  the  geocentric 
position  of  the  moon  was  as  follows: 

Zen4th  distance  =  z  =  65°  4O;  46".  5; 
Azimuth  =  a  —  48°  19'  49/x.8. 

Required  the  parallax  in  azimuth  and  zenith  distance  for 
Bethlehem. 

We  have  found  for  the  latitude  of  Bethlehem  (Art.  77) 

cp  —  cpf  =  ii'  22".  19; 
log  p  =  9.9993875- 

From  the  Nautical  Almanac,  page  113, 

TT  =  56'  2o".4. 

Our  computation  is  now  as  follows: 


IJ8 


PRACTICAL  ASTRONOMY. 


§81 


a   =  48°  19'  49". 8 
<p  —  qJ  =         n'  22". 19 
z   =  65°  40'  46". 5 

y    —  7'  33". 54 

it   —         56'  2o".4 
2  -  y    =  65°  33'  I2".g6 


*   =  8".  145 

5'  =  23'  I6".Q2 

45°+ iS-   =45°  oo'    4".<>7 


cos  a  =  9.8227125 

sin  =  7.5194794 

cosec  =  .0403593 

logp  =  9.9993875 
sin  it  =  8.2145238 

cos  =  9.6168344 
cos  (<p  —  <p')  =  9.  9999976 
—  .0000009 


cos  a  =  9.8227125 
log(<p  -  <?')  =  2.8339053 

log  y  =  2.6566178 


sin  3  =  5.5964625 
sin  3'  =  7.8307442 


tan  a  =  0.0506037 

tan  3  =  5.5964625 

tan  (45°  +  -J3)  =  .0000171 


tan  (a1  —  a)  —  5.6470833 
(a1  -  a)  =        9". 152 

tan  3'  —  7.8307540 

tan  (45°  +  i2')  =      0029412 

tan  (z  —  y)=    .3423734 

tan  (z'  —  z)  =  8.1760686 
(*'  -  g)=5i/33//.58 

a'  =  48*  19'  59". o 
z'  =  66°  32'  20". I 


We  thus  have  for  the  apparent  place 

Take  the  following  example  of  application  of  (VI),: 


log  p  =  9.9993875 
sin  it  —  8.2138035 
cos  rz  _^  _  ^)]  -  9.7995903 


sin  S'  =  8.0127813 


Zenith  distance  of  moon  \  z=  51°  06'  45".$ 
at  culmination,  f 

Equatorialhorizontalpar-  \n=        *&  14". 8 

allax,  ) 

cp  —  (p'  =        n'  22".  19      tan  [0  —  (cp  —  g>!)]  =  0.0904399 

<y  __        35/  24". 29  tan  3r  =  8.0128043 

45°  +  IS'  =  45°  17',  42".  15          tan  (45°  +  1^')  =    -0044727 

tan  (a/  —  z)  =  8.1077169 


§  82.     PARALLAX  IN  AZIMUTH  AND  ZENITH  DISTANCE.     1  39 

Case  Second. 

82.  To  compute  the  parallax  in  azimuth  and  zenith  dis- 
tance, having  given  the  observed  azimuth  and  zenith  distance. 

To  obtain  the  expression  for  (zr  —  z)  we  multiply  the  first 
of  (154)  by  cos  z'  and  the  second  by  sin  z',  and  subtract.  We 
thus  have 

sin  ""  cos    >~    ">  sin  (z'  -  A  .   (,64) 


For  (af  --a)  we  multiply  the  first  of  (148)  by  sin  a  ',  the 
second  by  cos  a'  and  subtract,  recollecting  that  cos  (a  —  a)=  i, 

sin  (a  —  a)  =  o.     We  thus  find 

t 

.     ,  ,        .        p  sin  7t  sin  (cp  —  cp'}  sin  a' 
sm  (a'  —a)  =  --  Y-  --  ^  --  .      .     .     .     (165) 

sin  z 

We  thus  have  for  the  parallax  in  zenith  distance  and  azi- 
muth, having  given  the  apparent  zenith  distance  and  azimuth, 


r  =  (9  —  9')  cos  a\ 

p  sin  n  cos  (cp  —  <z/)  sin  (z'  —  y] 

sin  (z   —  z)  = J 1—L — ;   I    /____, 

cos  y  \  (VII) 

,  ,  p  sin  n  sin  (cp  —  cp'\  sin  a' 

sm  (a  —  a]  = ^ . 

sin  z 

To  compute  y  we  may  substitute  a'  for  a  without  appre- 
ciable error. 

To  compute  (a1  —  a)  we  must  first  obtain  z  by  applying 
the  correction  (z'  —  z)  to  the  observed  zenith  distance. 

In  the  meridian,  a  =  a'=  o,  whence  y  —  cp—  cp' ,  a'  —  a  =  o, 
and  (VII)  become 

sin  (zr  —  z}  =  p  sin  n  sin  [zf  —  (cp  —  cp')~\  .     (VII), 

For  all  bodies  except  the  moon  (VII)  may  be  greatly  sim- 
plified, as  follows: 


140 


PR  A  C  TIC  A  L   ASTR  ONOM  Y. 


§82. 


(zr  —  z\  (a!  —  a),  and  n  being  very  small,  we  may  write 
the  arcs  in  place  of  their  sines.  (<p  —  g>')  and  y  being  small, 
we  may  write  for  their  cosines  unity.  We  then  have 

y  —  (cp  —  cp')  cos  a-,  \ 

z'  -  z  =  np  sin  (*'  --7);  (      (VIII) 

a'  —  a  =  Trp  sin  (<p  —  <p')  sin  a'  cosec  z.  } 

In  computing  these  we  may  use  a  and  z  or  a'  and  z'  in- 
differently in  the  second  terms.  It  will  often  be  sufficiently 
accurate  to  use 


=  7r 

a'  —  a  —  o. 


These  last  are  what  we  obtained  when  we   treated  the 
earth  as  a  sphere. 


Application  of  Formula  (VII). 

Latitude  of  Bethlehem  =  cp  =  40° 
Apparent  azimuth  of   moon  =  a'  —  48° 
Apparent  zenith  distance  of  moon  =  z'  =  66° 
Equatorial  horizontal  parallax  =  it  — 
cp  —  <pf  = 

\og(cp  -  cp'}  =  2.8339053 
cos  a'  =  9.8226904 


cos  (<p  —  cp')  = 
sin  (z'  —  y)  — 
sec       = 


=  2.6565957 
r  =      453"-52 
-  y  =  66°  24'  46".  58 


z'-z=         5i'33".58 
2  =  65°  40'  46".52 

a,'  —  a  =       ,          9".  152 
a  —  48°  19'  49". 85 


36'  23". 9 
19'  59"-o 

32'  20".  I 
56'   20".4 
II'   22". 19 
9.9999976 
9.9621103 
.0000009 


log/3  = 

sin  Tt  = 


9.9993875 
8.2145238 


sin  (cp  —  (p1)  —  7.5194794 
sin  a'  =  9-8733333 
.0403593 


cosec  z  = 


sin  (z1  —  z)  =  8.1760201 
sin  (a!  —  a)  =  5.6470833 


§  82.     PARALLAX  IN  AZIMUTH  AND  ZENITH  DISTANCE.      141 

Application  o 


Apparent  zenith  distance  of  the  moon  J  _    ,  _  „«  „„<>    „,/  A 

at  meridian  passage  f  = 

Equatorial  horizontal  parallax  =  TT  =          56'  14".  8 

<p  —  <p'  = 

logp  =  9,9993875 

sin  7t  =  8.2138035 

sin  [«'  —  (<p  —q>')]  =  9.8944903 

sin  (2'  —  z)  =  8.1076813 
a'  —  2  =    44'  3".  13 


Application  of  (VIII). 

Find  the  parallax  in  azimuth  and  zenith  distance  of  Venus 
as  seen  from  Bethlehem,  having  given  the  following: 

a  =  271°  56'  21"  log  (<p  —  <?')  =  2.83390  sin  (z  —  y}  =  9.96312 

z=    66°  43'  35"  cos  «=  8.  52941 

it  =                 13".  61  log  p  =  9.99939 

---  log^  =  1.36331  log  it  —  1,13386 


7  = 


—  y=    66°  43' 12"  sin  (<p  —  (p')  =  7-51947 

sin  a  =  9  99975n 
cosec  z  =    .03686 


z1  —  z  =  +  12". 48         log  (z1  —  z)  =  1.09637 
a'  —  a  —  —      ".05         log  (a!  —  a)  =  8.68933,, 


For  this  case  formula  (VIIIj)  gives 

log  7t  =    I.I3386 

sin  z  =  9.96314 


log  (z'  —  z)  =  1.09700         z'  —  z  =  -f-  12".  50 


PRACTICAL  ASTRONOMY.  §  83. 


Application  of  (VII),. 

Zenith  distance  of  Venus  at  time  of  transit  =  z  =  24°  15'  35" 
Equatorial  horizontal  parallax  =  it  —  13".  57 

q>  —  q><  =          u'  22" 
log  7T  =  1.13258 
log  p  =  9.99939 
sin  [z  —  (<p  —  tp')]  =  9.61051 

log  (*'  -  z)  =    .74248        z'  —  z  =  5".  53 


Third. 

83.  Required  the  parallax  in  right  ascension  and  declina- 
tion, having  given  the  geocentric  right  ascension  and  declina- 
tion. 

Let  the  equator  of  the  observer  be  taken  as  the  plane  of 
x,  y,  the  positive  axis  of  x  being  directed  to  the  vernal  equi- 
nox, the  positive  axis  of  y  to  that  point  on  the  equator  whose 
right  ascension  is  90°,  and  the  positive  axis  of  z  to  the  north 
pole  of  the  heavens. 

Let         x' ,  y,  z'  =  the  rectangular  co-ordinates ; 
A' i  a',  $'  =  the  polar  co-ordinates. 

We  then  have          x'  =  A'  cos  8'  cos  a' ;  \ 

y  =  A'  cos  tf'  sin  of  ;  V  .  .     .     .     .     (166) 
*'  =  A'  sin  <*'.  ) 

In  the  second  system  let  the  origin  be  at  the  centre  of  the 
earth,  the  axes  being  respectively  parallel  to  those  of  the 
first  system. 

Let          x,  y,  z  be  the  rectangular  co-ordinates ; 
J,  OL,  d  be  the  polar  co-ordinates. 


§  83.   PARALLAX  IN  XT.  ASCENSION  AND  DECLINATION.    143 

Then  x  =  4  cos  3  cos  a ;  \ 

y  =  A  COS  8  sin  a  ;  >- (l&7) 

z  —  A  sin  <?.  ) 

Let  now 

•*.».7o»*o  =  rectangular  co-ordinates  j  of  the  observer's  position 

r      referred  to  the  eaith's 
/o,  <£/,  0  =  polar  co-ordinates  centre. 

Here  p  is,  as  before,  the  line  joining  the  observer's  position 
with  the  centre  of  the  earth,  and  <p'  and  0  are  respectively 
the  declination  and  right  ascension  of  the  point  where  this 
line  produced  pierces  the  celestial  sphere  ;  or  in  other  words, 
of  the  geocentric  zenith.  The  declination  of  the  zenith,  as 
we  have  seen  (Art.  63),  is  equal  to  the  latitude  —  <p'  in  this 
case. 

The  right  ascension  of  the  zenith,  0,  equals  the  right  ascen- 
sion of  the  observer's  meridian — all  points  on  the  same  me- 
ridian having  the  same  right  ascension.  This  we  shall  see 
hereafter  is  equal  to  the  observer's  sidereal  time. 

We  have  then      XQ  =  p  cos  <pf  cos  6  ;  \ 

y^  —  p  cos  (pf  sin  8  ;  v 068) 

#0  =  p  sin  cpf ;  ) 

and  for  passing  from  system  (166)  to  (167), 

**  =  x  -  x, ;        y'  =  y  -  y, ;        *>  =  *  —  *,.      (169) 
Therefore 

A'  cos  8'  cos  a'  =  A  cos  3  cos  a  —  p  cos  q>'  cos  & ;  ) 

A'  cos  $'  sin  a'  =  A  cos  8  sin  a  —  p  cos  <p'  sin  0 ;  V  (170) 

A'  sin  £'  —  J  sin  £  —  p  sin  ?>'.  ) 


144  PRACTICAL   ASTRONOMY.  §  84. 

As  before,  let  us  divide  through  by  Ay  and  write 

A'  i 

/=--;         sin*=-. 

Then 

f  cos  6'  cos  «'  =  cos  d  cos  a  —  p  sin  TT  cos  <p'  cos  6  ;  j 
/cos  £'  sin  a'  —  cos  tf  sin  a  —  p  sin  TT  cos  cp'  sin  6?-;  v  (171) 
/  sin  6'  =  sin  $  —  p  sin  n  sin  £>'.  ) 

Let  us  diminish  all  horizontal  angles  by  <*,  which  will 
be  equivalent  to  transforming  our  rectilinear  systems  to 
others  in  which  the  axes  of  x  and  x1  make  respectively  the 
angle  a  with  the  original  axes.  We  thus  derive 

/cos  dr  cos  (a  —  a)  —cos  d—  p  sin  n  cos  cp'  cos  (0  —  -  or);  )  ,       . 
/cos  d'  sin  (^  —  «)  =  —  p  sin  ?r  cos  ^  sin  (0  —  a).  }  (- 

p  sin  n  cos  <z/ 
Let  us  write  «'  =  --  —  -^  --  ,    .....     (t/3) 

which  substituted  in  (172)  and  the  second  divided  by  the 
first,  we  find     , 

m  sin  (a  —  0) 
tan   «'  -  «   =  __7_.    .    .    (174) 


84.  As  in  case  first,  we  may  give  this  a  form  better  adapted 
to  logarithmic  computation,  as  follows:     Write 

p  sin  TT  cos  q>f  cos  (or  —  0) 
sin  5  =  m1  cos  (a  -  0)  -  -  -  -  •      (i/5) 


Then  (174)  becomes 

tan  (a'  -<x)  =  tan  (a  -  0)  -~_  sin 


§  84.    PARALLAX  IN  RT.  ASCENSION  AND  DECLINA  TION.    14$ 

But 

sin  S      _  __  sin  5 
i  -  sin  5  "~  cos'2  1$  —  2  sin  $$  cos 
sin  $ 


(cos  p  -  sii 

sin  5(cos  f9-  -f  sin 


__ 

(cos  J3+  sin  £3)  (cos  JS—  sin^S)  (cos  £2-  sin  p) 
sin  3"  cos  JS"       sin  ^3- 


cos  JS"    -  sin 

=  tan  3  tan  (45°  + 


Therefore 

tan  (a'  -  a)  =  tan  (or  -  0)  tan  3  tan  (45°  +  JS),    (176) 

which  determines  («7  —  or).  For  determining  (£'  -  -  6)  we 
multiply  the  first  of  (172)  by  cos  £(<*'  --  a),  the  second  by 
sin  £(«'  —  a);  add  the  products,  and  divide  the  result  by 
cos  ^(V  —  or).  By  this  process  we  obtain 


,  cos  \Maf  +  '<*)  —  01 

/  cos  o'  —  cos  tf  —  p  sin  n  cos  <p TTTEJ — — \ — -• 

COS  |-(V  —  or) 

The  last  of  (171)  is 

f  sin  #'  =  sin  8  —  p  sin  TT  sin  (pf. 
Let  us  write 


J77) 


tan  cpf  cos  £(«'  —  ex) 

tan  y  = .T..   ,    .      x ~  ....     (178) 

cos  [•£(<*  -f  a)  —  0] 

Then  (177)  become 

j  sin  tf'  =  sin  d  -  p  sin  it  sin  ^  ;  )  ,       , 

y  cos  tf'  —  cos  §  —  p  sin  n  sin  <?/  cot  7.  f 


146  PRACTICAL  ASTRONOMY.  .      §  84. 

Multiply  the  first  of  these  by  cos  #,  the  second  by  sin  #,  and 
subtract  ;  then  multiply  the  first  by  sin  #,  the  second  by 
cos  #,  and  add.  We  thus  obtain 


/sin  (6'  -  d)  =  p  sin  n  sin  <p' 


sin  y 
/cos  (#'  -  d)  =  I  -  p  sin  n  sin  <p'  -     gin  ~  Y\ 


p  sin  n  sin  <p  , 

Let  us  write          n'  = (181) 


Introducing  this  value  and  dividing  the  first  equation  by  the 
second,  we  find 


Then  writing 

p  sin  n  sin  <pf  cos  (S  —  y) 
sin  y  -  tf'cos  (d  -  y)  =  £  gm  ^  H   (182) 

this  equation  becomes 

tan  <*'  -  5)  =  Sln  f  ^"inV  ^  =  ta"  (^  "  X)  ta°  ^  ta"  (45°  +  ^'   (l83) 

Equations  (175),  (176),  (178),  (182),  and  (183)  give  the  com- 
plete solution  of  the  problem. 

We  thus  have  for  computing  the  parallax  in  right  ascen- 
sion and  declination,  having  given  the  geocentric  right  ascen- 
sion and  declination,  the  following  formulas: 


g  84.    PARALLAX  IN  RT.  ASCENSION  AND  DECLINATION.     147 

p  sin  n  cos  q>'  cos  (8  —  a) 


sin  3-  = 


cos 


tan  (a  —  a')  =  tan  (8  —  a)  tan  3  tan  (45°  + 
tan  <p'  cos  \(a  —  a'} 


tan  y  — 


sin  sr  = 


cos 


+  a')  — 


sin  TT  sin  <px  cos  (y  — 


sn 


tan  (tf  -  <?')  =  tan  (7  -  d)  tan  3'  tan  (45°  +  JS').  . 


In  the  meridian,  a:  =  a'  =  6.     Therefore  y  =  (p',  and  the 
above  become 


sin  3-'  =  p  sin  n  cos  (^  —  tf); 
tan  (d  -  (T)  =  tan  (^  -  tf)  tan  5r  tan  (45°  + 


K       (IX), 


Application  of  Formula  (IX). 

Required  the  parallax  of  the  moon  in  right  ascension  and 
declination,  1881,  July  4th,  Qh,  Bethlehem  mean  time,  as  seen 
from  Bethlehem. 

Converting  9h  mean  time  into  sidereal  time  by  the  method 
to  be  explained  hereafter  (p.  170),  we  have 

Bethlehem  sidereal  time  =  0  =  I5h  52™  5O8.2 

From  the  Nautical  Almanac,  p.  114,  we  find  a  =  i2h  57™  io8.s6 

S  —  —  n°     3'  48". 4 

Astronomical  latitude  of  Bethlehem  =  (p  =  40°  36'  23". 9 

<p  —  qj  —  n'  22". 2 

Geocentric  latitude  of  Bethlehem  =  <p'  =  40°  25'    i'  .7 

Nautical  Almanac,  p.  113,  equatorial  horizontal  parallax  =  it  =  56'  20". 4 

6  —  a  =  2h  55m  398.64 

=  43°  54'  54". 6 


148 


PRACTICAL   ASTRONOMY. 


§84. 


cos  (0  -  or)  =  9.8575542 
sec  8  =    .0081471 


COS  <f> 

=  9.8815812 

tan  <p' 

=  9.9302268 

log/3 

=  9.9993875 

cos  i(a  —  a') 

=  9-9999957 

sin  it 

=  8.2145238        sec 

[}(a  +  O-  6] 

=    .1443121 

sin  <p' 

=  9.8118080 

tan  y 

=     .0745346 

cos  (^  —  8) 

=  9.6861710 

r 

=  49°  53'  33". 

56 

cosec  x 

=    .1164301 

y-3 

=  60°   57'  2l". 

96 



tan  (^  -  6) 

=  0.2554636 

sin  5 

=  7.9611938 

tan  $' 

=   7.8283302 

3- 

=         31'  26".36 

tan  (45°  +  |3') 

=      .0029250 

45°  +  *S 

A  e°  re'  A  a"  *> 

—  45     L5   43   -^ 

•  •   ft  nRA-TTSR 

sin  3-' 

=  7.8283204 

t3-Il  ^O           O  ^ 

=  4i'  58".39 

3' 

=    o°  23'    9".  15 

tan  (0  -  a) 

=  9-9835502 

45°  +  i*' 

=  45°  n'  34".  6 

tan  3 

=  7.9612118 

tan  (45°  +  i$) 

=    .0039719 

tan  (a  —  a') 

=  7.948^339 

or-  a' 

=      o°  30'  32 

".94 

a  —  194°  17'  38".4 

therefore  a'  =  193°  47'    5".  5 

i(a+«')  =  194°    2'2i".9 

0  =  238°  12'  33".o 

'  +  a)-B  =  315°  49'  48".9 


We  therefore  have  for  the  position  of  the  moon  as  seen 
from  Bethlehem,  1881,  July  4th,  9h,  mean  time, 


8S.36; 


=  -  11°  45; 


Application  of  (IX,). 
At  the  time  of  meridian  passage  at  Bethlehem,  1881,  July 


g  85.    PARALLAX  IN  RT.  ASCENSION  AND  DECLINA  TION.     149 


4th,  the  moon's  declination  and  equatorial  horizontal  paral- 
lax were  as  follows: 


d  =  -  10°  30'  21". 6 
n  =  56'  -I4".8 

<p'  =       40°  25'    i".; 
-#=       50°  55'  23".3 

log  p  =  9.9993875 

sin  Tt  —  8.2138035 

cos  (9?'  —  d)  =  9. 79959°3 

sin  3'  =  8.0127813 

3'  =          35'  24". 29 
45°+**'  =  45°  17' 42".  i 


Required  (6  — 


tan  (<?'  —  d)  =    .0904399 

tan  3'  =  8.0128043 
tan  (45°  +  |3)  =    .0044726 


tan  (d  —  d')  =  8.1077168 
S  -d'  =  44'  3".  13 


Case  Fourth. 

85.  Required  the  parallax  in  right  ascension  and  declina- 
tion, having  given  the  apparent  right  ascension  and  declina- 
tion. 

Multiply  the  first  of  (171)  by  sin  <*',  the  second  by  cos  oc'; 
subtract  and  reduce.  The  result  is 

p  sin  7t  cos  q>'  sin  (0  —  a') 
Sm   "-"=  -~  -'    ' 


To  obtain  d  —  d'  we  make  use  of  (179).  Multiply  the 
first  by  cos  8  ',  the  second  by  sin  d';  subtract  and  reduce.  We 
thus  have 

P  sin  *  sin       sin  fr  -  .T)          ( 


sn       _        = 


sin 


We  have  therefore  the  following  formulae  for  the  parallax 
in  right  ascension  and  declination,  having  given  the  appar- 
ent co-ordinates: 


ISO 


PRACTICAL   ASTRONOMY. 


§85. 


.     ,  p  sm  n  cos  cp  sin  (9  —  a'} 

sm  (a  —  a'}  =  - —  £— - — i '-• 

cos  6 

tan  a/  cos  4(a  —  a'\ 

tanr:=cos[K«  +  «')-e]; 

p  sin  TT  sin  <z/  sin  (v  —  d'\ 
sm  (tf  —  <?')  =  - — 


sn 


(X) 


To  compute  the  first  of  these  we  require  tf,  which  will 
be  unknown  until  after  we  have  computed  the  last,  which 
in  turn  requires  a  knowledge  of  a  obtained  from  the  first. 
We  must  therefore  proceed  indirectly  as  follows:  Compute 
(a—  a'\  using  in  the  denominator  8'  instead  of  #.  With  the  ap- 
proximate value  of  a  so  obtained  compute  (tf  —  $')•  this  gives 
us  tf,  with  which  we  recompute  (a  —  a').  It  will  never  be 
necessary  to  repeat  the  computation  of  d  —  d'  with  this  new 
value  of  a. 

In  the  meridian,  a  =  a'  —  6.  Therefore  y  —  (p ',  and 
formulae  (X)  become 

sin  (d  —  #')  =  p  sin  n  sin  (cpr  —  6'). .     .     .     (X)t 

For  all  bodies  except  the  moon  we  may  write,  without 
appreciable  error, 


sn  TT  = 


sn      - 
cos 


cos      «-«= 


=  cos 


giving  the  following  approximate  formulae: 
f        np  cos  q>'  sin  (0  —  a'} 

"-<*=-          cos<r        "' 


tan  y  = 

O    —    O    = 


tan  (?>' 

cos  »  -  «o; 

y7  sin  (^  — 


sm  y 


•    •    (XI) 


§  85.    PARALLAX  IN  RT.  ASCENSION  AND  DECLINATION.    1 5 1 

In  these  formulae  we  may  use  either  the  geocentric  co- 
ordinates (a  and  #)  or  the  observed  (a  and  d')  indifferently. 
In  the  meridian,  where  6  =  a  =  a',  y  =  <pr  and  (XI)  become 

8  -  6'  =  np  sin  (<pf  -  <T).    .     .     .     .     (XI), 

Application  of  (X). 

Required  the  geocentric  place  of  the  moon,  having  given 
the  apparent  place  as  seen  from  Bethlehem,  1881,  July  4th, 
9h,  Bethlehem  mean  time,  as  follows: 

Apparent  right  ascension  =  OL   =        I2h  55m    8s. 36; 
Apparent  declination  =  d'  =  —  11°  45'    46". 79. 

From  Nautical  Almanac,  p.  113,  it  —  56'    20". 4 

Geocentric  latitude,  cp'  =       40°  25'      i".7 

Sidereal  time,  0  =        I5h  52™  so8 .2 

0  —  a'  =       44°  25'   27". 6 
sec  d'  =    .009  2176 

*sec  d  =    .008  1471  Approx.  sin  (a  —  a'}  =  7.9497875 

cos  q>  =  9.881  5812  Approx.  (a  —  a')  =  30'  37". 5 

sin  (9  -  a')  =  9.845  0774  a'  =  193°  47'    s".4 

Approx.  a  =  194°  17'  43 

log  p  =  9.999  3875  |(a  -j-  a')  =  194°    2'  24". 2 

sin  TT  =  8.214  5238  [i(a  -j-  a')  —  6]  =  315°  49'  5i".2 

-Ka  -  a')  =  15'  i8".8 


sin  cp'  =  9.811  8080  

sin  (y  —  8')  =  9.944  5358  tan  cp'  =  9.9302268 

cosec  Y  —    •II^  4320  cos^(or  —  a')  =.  9.9999957 

sec  |>(a:  +  a')  -  6]  =    .1443074 
sin  (d  -  d')  =  8.086  6871  


6  —  8'  =  41'  58". 39  tan  y  =    .0745299 

8  =  -  11°     3'  48".4  ^  =  49°  53'  32".5 

y  —  8'  =  61°  39'  19".$ 
Corrected  sin  (a  —  a')  =  7.9487170 

True  (a  —  a')  =          30'  32".  94 
a  =  194°  17'    38".34 
_    I2h  57m  io8.55 

This  value  is  inserted  after  the  computation  of  the  parallax  in  declination. 


152  PRACTICAL  ASTRONOMY.  §8$. 


Application  of  (X),. 

1 88 1,  July  4th,  at  meridian  passage,  Bethlehem,  the  moon's 
apparent  declination  and  equatorial  horizontal  parallax  were 
as  follows : 

£'  =  —  u°  14'  24".7   Required  the  parallax  in  declination. 


=       40°  25' 


9/-cT  =        51°  &'  26". 


log  p  =  9-9993875 

sin  TT  —  8.2138035 

sin  (<p  —  d')  =  9.8944903 

sin  (d  —  d')  =  8.1076813         5  —  d'  =  44'  3".  13 


Application  of  (XI). 

1 88 1,  July  4th,  i6h,  Bethlehem  mean  time,  the  right  ascen- 
sion, declination,  and  equatorial  horizontal  parallax  of  Venus 
were  as  follows: 

From  Nautical  Almanac,  p.  355,  a  =    3h  46™  12s  .25 

d  =  16°  18'  23".3 

From  Nautical  Almanac,  p.  388,  it  -—  13' '.61 

Sidereal  time,*  8  =  22h  53m  59s  .2 


See  p.  170. 


86. 


REFRACTION. 


The  computation  is  then  as  follows  : 


0  —  a  =    i9h    7m  47s 

=  286°  56'   45" 
tp'  =    40°  25'      i".7 
y  =    71°    6'    27" 
y  -  d  =    54°  48'     4" 


cos  (0  —  cz')  =  9.46459 
tan  0>'  =  9.93027 

tan  y  =    .46568 


cos  <p'  —  9.88156 
sin  (6  —  a)  —  9.98072,1 
sec  5  =    .01783 

log  p  -  9.99939 

log  7C  •=.    I   13386 


sin  (y  —  d)  =  9  91231 

sin  q>'  =  9  81183 

cosec  JK  =      02405 

log  («r  —  a')  =  i.oi336» 
log  (d  -  5')  =    .88144 


Application  of 

To  compute  the  parallax  of  Venus  in  declination  at  the 
time  of  meridian  passage,  Bethlehem,  1881,  July  4th. 
The  data  are  as  follows : 


a  —  a'  =  —  io".3i 

=  -  '.69 

d  -  d'  =  +    7".6i 


=  1  6°  20' 


i3"-57 


5"-53 


7T   —    I.I3258 

log  p  =  9.99939 
sin  (cpr  --  6)  =  9.61051 

log  (d  -  V)  =±    .74248 


Refraction. 

86.  When  a  ray  of  light  passes  obliquely  from  a  rarer 
into  a  denser  medium,  it  is  bent  or  refracted  out  of  its  origi- 
nal course  towards  the  normal  drawn  to  the  surface  separating 
the  two  media,  at  the  point  where  the  ray  pierces  this  surface. 
The  angle  which  the  original  direction  of  the  ray  makes  with 
this  normal  is  the  angle  of  incidence,  and  the  angle  formed  with 
the  normal  by  the  bent  or  refracted  ray  is  the  angle  of  refrac- 
tion. 


154  PRACTICAL  ASTRONOMY.  §86. 

According  to  Descartes,  refraction  takes  place  in  accord- 
ance with  the  following  laws  : 

I.  Whatever  the  obliquity  of  the  incident  ray,  the  ratio  which  the 

sine  of  the  angle  of  incidence  bears  to  the  sine  of  the  angle  of 

refraction  is  always  constant  for  the  same  two  media,  but 

varies  with  different  media. 

II.  The  incident  and  refracted  ray  are  in  the  same  plane,  which 

is  perpendicular  to  the  surface  separating  the  two  media. 
If  the  density  of  the  air  were  uniform  and  constant,  the 
determination  of  the  effect  of  refraction  would  be  a  com- 
paratively easy  matter  in  accordance  with  these  laws.  Neither 
condition  is  realized,  however. 

The  density  of  the  air  is  a  maximum  at  the  surface  of  the 
earth,  and  it  continually  decreases  as  we  ascend  above  the 
surface,  until  it  practically  disappears  at  an  altitude  of  45  or 
50  miles.  It  is  also  continually  varying  in  density,  as  shown 
by  the  readings  of  the  barometer  and  thermometer. 

In  consequence  of  the  decrease  in  density  of  the  air  as  we 
ascend  above,  the  surface  of  the  earth,  it  follows  that  the 

path  of  a  ray  of  light  through 
the  atmosphere  is  not  a  straight 
line,  but  a  curve,  as  shown  in 
the  figure.  We  see  a  star  in 
the  direction  of  a  tangent 
*  drawn  to  the  curve  at  the 
point  where  it  enters  the  eye. 
In  consequence,  the  altitudes 
of  all  celestial  bodies  appear 
to  us  greater  than  they  really 
are  ;  but  in  accordance  with 
Descartes'  second  law,  the  azi- 
FlG  8  muths  are  not  affected  at  all. 

It  sometimes  happens  that  there  are  lateral  deviations  of 
an  anomalous  character,  but  these  are  beyond  the  scope  of 


§86.  REFRACTION.  155 

theory,  and  when  they  exist  are  generally  to  be  counted 
among  the  accidental  errors  to  which  observations  are  liable. 

The  complete  investigation  of  the  laws  of  astronomical  re- 
fraction is  a  very  complex  and  difficult  problem,  and  one 
which  has  never  been  solved  with  entire  satisfaction.  We 
shall  not  enter  into  the  theory  here,  but  confine  ourselves  to 
the  explanation  of  the  use  of  our  refraction  tables  based  on 
those  of  Bessel. 

Bessel's  formula  for  the  amount  of  refraction  at  any  zenith 
distance  z  is 

r  —  a./3Ay*  tan  z (186) 

In  which  r  is  the  refraction ;  a  varies  slowly  with  the  zenith 
distance ;  A  and  A  also  vary  with  the  zenith  distance,  and 
differ  but  little  from  unity.  This  difference  is  never  appre- 
ciable except  for  large  zenith  distances :  for  our  purposes  it 
will  generally  be  sufficiently  accurate  to  regard  them  as 
unity.  /3  is  a  factor  depending  on  the  barometer  reading. 
As  this  reading  depends  on  the  pres'sure  of  the  air  and  the 
temperature  of  the  mercury,  it  is  tabulated  in  the  form 

ft  =  t  X  B. 

In  which  B  depends  on  the  reading  of  the  barometer,  and  / 
upon  the  attached  thermometer. 

y  depends  upon  the  temperature  of  the  air  as  shown  by  the 
detached  thermometer. 
We  may  therefore  use  the  formula 

r  =  R  X  B  X  t  XT. (187) 

In  which  R  =  a  tan  z  is  given  in  table  II  A; 

.5  depends  upon  the  barometer  and  is  given  in  table  II  B; 

t  depends  upon  the  attached  thermometer  and  is  given  in  table  II  C; 

T  depends  upon  the  detached  thermometer  and  is  given  in  table  II  D. 


I  5  6  PRA  C  TIC  A  L  A  S  Tit  ONOM  Y.  §  86. 

As  an  example  take  the  following: 

Apparent  altitude  =  h  =  31°  49'  48" 
Barometer  reading  29-Sl  inches 

Attached  thermometer        78°.2 
Detached  thermometer       82°.  i 

Table  II  A,  R  =  95".  6  log  =  1.9713 

II  B,  B  =  .983  log  =  9.9928 

II  C,  t  =  .997  log  =  9.9990 

II  D,  T  =  .941  log  = 


r  =  i'  26".  4  \ogr=  1.9367 

For  many  purposes,  especially  for  small  zenith  distances, 
it  will  be  sufficiently  accurate  to  use  the  mean  refraction  R 
without  correcting  for  barometer  and  thermometer. 

An  approximate  value  may  be  obtained  by  the  formula 

r  =  57".7  tan  z  .....     .     .     (188) 

This  will  be  accurate  enough  for  many  purposes,  and  may 
be  of  service  in  cases  where  tables  are  not  available.  This 
would  give  for  our  example  above 

r  =  i'  32".95- 

When  the  greatest  precision  is  demanded,  table  III  must 
be  employed.  For  the  above  example  we  have 

<• 

Table  III  A,  log  a  =        1.76021  A  =       i.oo  A  =  1.004 


111  B'[  A.  log  ft  =-  .00306  lo^=  -°°127 

III  C,  )  log    /  =  -  .00179 

III  D,       A.  .  log  y  =  —  .02757  log  Y  =  —  .02746 

tan  z  =  .20709 


r  =  i'  26". 43  log  r  =        1.93667 


§87.  REFRACTION.  157 

In  the  volume  of  astronomical  observations  of  the  Wash- 
ington Observatory  for  1845  may  be  found  refraction  tables 
carried  out  much  farther  than  those  given  here.  They  are 
convenient  when  many  computations  are  to  made  with  great 
precision. 

Refraction  in  Right  Ascension  and  Declination. 

87.  As  our  tables  give  the  refraction  in  zenith  distance  or 
altitude,  if  we  require  the  effect  in  right  ascension  and  decli- 
nation it  will  be  necessary  to  express  the  increments  of  these 
quantities  in  terms  of  the  increment  of  the  zenith  distance. 
Differential  formulas  will  be  accurate  enough  for  any  case 
which  is  likely  to  arise.  Such  formulae  are  given  in  works 
on  Trigonometry.  Those  required  for  this  particular  pur- 
pose are  derived  as  follows : 

Let  us  assume  the  general  formulae  of  spherical  trigonome- 
try, viz.: 

cos  a  =  cos  b  cos  c  -|-  sin  b  sin  c  cos  A;  \ 
sin  a  cos  B  —  cos  b  sin  c  —  cose  sin  b  cos  A\  (  .    (189) 
sin  a  sin  B  —  sin  b  sin  A.  ) 

Applying  these  formulas  to  the  triangle  formed  by  the  zenith, 
the  pole,  and  the  star,  we  have 

sin  tf=sin  q>  cos  z— cos  <p  sin  z  cos  a; 
cos  8  cos  ^=sin  <p  sin  ^4~cos  9>  cos  z  cos  a\ 
cos  #  sin  ^—  cos  9  sin  a. 

Also, 

cos  2= sin  q>  sin  tf-|-cos  <p  cos  8  cos  /; 
sin  z  cos  <7=sin  <p  cos  d— cos  q>  sin  d  cos  /; 
sin  2  sin  ^—  cos  <£>  sin  t.  FIG.  9. 


158.  PR  A  C  TIC  A  LAS  TRONOM  Y.  §  S/. 

Now  differentiating  the  first  of  (190),  regarding  6  and  zonly 
as  variables, 


cos  ddd  =  —  (sin  q>  sin  z  -j-  cos  q>  cos  z  cos  a)dz. 
Combining  this  with  the  second  of  (190),  we  have 

dd  =  —  cos  qdz  .......     (r92) 

Differentiating  the    first  of  (191),  regarding  z,  6,  and  t  as 
variables, 

—  sin  ££&:=:  (sin  cp  cos  #—  cos  cp  sin  d  cos  t)dd—  cos  9  cos  tf  sin  tefr. 

Combining  this  with  the  second  and  third  of  (191)  and  with 
(192),  we  readily  derive 

cos  ddt  =  +  sin  qdz  ......     (193) 

In  (192)  and  (193), 

dz  =  the  refraction  in  zenith  distance  =  r\ 
*t  =  @  —  a\        therefore        dt  =  —  da. 

Our  formulae  then  become 

dd  = 


I:  :••;«?  •  •  •  • 


cos      *  =  —  r  sn  q, 

For  applying  these  formulas  we  must  compute  ^,  and  we 
require  z  for  taking  from  the  table  the  refraction  in  zenith 
distance. 

Equations  (191)  give  these  quantities,  the  solution  of  which 
is  a-s  follows : 

Let  n  sin  ^V  =  cos  q>  cos  t\ 

n  cos  N  =  sin  (p. 


§87- 
Then 

and  finally, 


REFRACTION. 

cos  z  =  n  sin  (d  -f- 
sin  ^  cos  q  —  n  cos  (A  -f-  A7"); 
sin  #  sin  ^  =  cos  9?  sin  /; 

tan  N  =  cot  <p  cos  /; 

sin  N 
tan  a  =  — 


tan  z  = 


*59 


cot  ($  +  . 
cos  ^ 
sin  N  cos  <p  cos 


(XII) 


cos  (d  -\-  N)        sin  ^  cos  q 

As  an  example  of  the  application  of  formulae  (194),  take 
the  following: 

Given  the  sun's  right  ascension  a  =       2ih  47m  59S.Q2 
Declination  tf  =  —  13°  if  '*$".'] 

Latitude  q>  =       40°  36'  24" 

Sidereal  time  ©  —         oh    om    os 

Barometer  reading 

Attached  thermometer 

Detached  thermometer 


29.5  inches 

65°.i 

;o°.o 


From  (XII)  we  find 
z  =  61°  58'.o;        cos  ^  =  9.94620; 
From  table  II  A,  R=  i'  49". o 


II  B, 
IIC, 
II  D, 


•983 
.998 
.962 


sin  q  =  9.67068. 

log  =  2.0374 
9.9927 
9.9994 
9.9834 


dd  =  -  9i.o 
cos  6 da  —  —     8", 


log  r  =  2.0129 
cos  q  =  9.9462 
sin  q  =  9.6707 

log  =  1.9591 
log  =  1.6836 


i6o 


PRACTICAL   ASTRONOMY. 


§88. 


Dip  of  the  Horizon. 

88.  At  sea,  altitudes  of  the  heavenly  bodies  are  measured 

from  the  visible  horizon,  which 
is  generally  a  clearly  defined 
line.  As  the  eye  of  the  observer 
H'  is  elevated  above  the  surface  of 
the  water,  this  visible  horizon, 
owing  to  the  curvature  of  the 
earth,  will  be  below  the  true 
horizon. 

Thus,  in  the  figure,  let  the 
circle  represent  a  section  of  the 
earth  made  by  a  vertical  plane 
passing  through  the  eye  of  the 
observer  at  A.  Then  AH  will  be  a  section  of  the  true  hori- 
zon; AC  will  be  a  section  of  the  visible  horizon;  the  dip  will 
be  the  angle  HAC  —  AOC. 

Let  D  =  the  dip; 

a  —  the  radius  of  the  earth  in  feet ; 

x  =  AB,  the  height  of  the  eye  above  the  water  in  feet. 

Then  from  the  triangle  A  CO, 


FIG. 


—  CO*  = 


or 


tan  D  — 


As  x*  will  be  very  small  in  comparison  with  2ax,  we  may 
neglect  it  without  appreciable  error.  Also,  D  being  a  small 
angle,  we  may  write 

tan  D  =  D  tan  i" '. 


§  88.  DIP    OF    THE  HORIZON.  l6l 


Therefore  we  have  D  = 


or  £>  =  63".82  Vx  in  feet (195) 

This  formula  would  give  us  the  true  value  of  tjie  correc- 
tion if  there  were  no  refraction,  the  effect  of  which  is  to  di- 
minish D.  The  refraction  very  near  the  horizon  is  always  a 
somewhat  uncertain  quantity,  but  for  a  mean  state  of  the 
air  the  dip  corrected  for  refraction  will  be  found  by  multi- 
plying the  value  given  by  (195)  by  the  factor  .9216, 


or  D"  =  58".82  Vx  in  feet (196) 

An  approximate  value  sometimes  used  by  navigators  is 
obtained  by  taking  the  square  root  of  the  number  of  feet 
above  the  water  and  calling  the  result  minutes.  Thus  if  the 
eye  is  25  feet  above  the  water,  this  process  would  give  for 
the  dip  5';  formula  (196)  gives  4'  54". 

The  dip  must  be  subtracted  from  the  observed  altitude  to 
obtain  the  true  altitude. 


CHAPTER  III. 

TIME. 

89.  For  astronomical  purposes  the  day  is  considered  as 
beginning  at  noon  instead  of  at  midnight;    the  hours  are 
reckoned  from  zero  to  twenty-four,  instead  of  from  zero  to 
twelve  as  in  civil  time.     Thus,  July  4th,  9h  A.M.,  civil  reckon- 
ing, would  be  July  3,  2ih,  astronomically. 

In  all  operations  of  practical  astronomy  the  time  when  an 
observation  is  made  is  a  very  important  element.  There  are 
various  methods  of  reckoning  time,  of  which  three  are  in 
common  use,  viz.,  mean  solar,  apparent  solar,  and  sidereal  time. 
Before  entering  upon  the  relations  between  these  different 
kinds  of  time,  some  preliminary  considerations  are  necessary. 

90.  The  transit,  culmination,  or  meridian  passage  of  a  heaven- 
ly body  at  any  place  is  its  passage  across  the  meridian  of 
that  place. 

Every  meridian  is  bisected  at  the  poles ;  and  as  a  star  in 
the  course  of  its  apparent  diurnal  revolution  crosses  both 
branches,  it  is  necessary  to  distinguish  between  the  upper 
culmination  and  lower  culmination. 

The  Upper  Culmination  of  a  heavenly  body  is  its  passage 
over  that  branch  of  the  meridian  which  contains  the  ob- 
server's zenith. 

The  Lower  Culmination  is  the  passage  over  that  branch 
which  contains  the  observer's  nadir. 

Any  star  whose  north-polar  distance  does  not  exceed  the 


§  90.  TIME,  163 

north  latitude  of  the  place  of  observation  is  constantly  above 
the  horizon,  and  may  be  observed  at  both  upper  and  lower 
culmination.  Any  star  whose  south-polar  distance  does  not 
exceed  the  north  latitude  of  the  place  of  observation  is  al- 
ways below  the  horizon,  and  therefore  cannot  be  observed 
at  all.*  Stars  between  these  limits  can  be  observed  at  upper 
culmination  only. 

The  rotation  of  the  earth  on  its  axis  being  uniform,  it 
follows  that  the  intervals  of  time  between  the  successive 
transits  of  a  point  on  the  equator  over  either  branch  of  the 
meridian  will  be  of  equal  length.  Such  an  interval  is  a  si- 
dereal day,  and  the  point  with  the  transit  of  which  the  side- 
real day  is  regarded  as  beginning  is  the  vernal  equinox. 

A  SIDEREAL  DAY  is  the  interval  between  two  successive  transits 
of  the  vernal  equinox  over  the  upper  branch  of  the  meridian. 

THE  SIDEREAL  TIME  at  any  meridian  is  the  hour-angle  of  the 
vernal  equinox  at  that  meridian. 

The  right  ascensions  being  reckoned  from  the  vernal  equi- 
nox, it  follows  that  a  star  whose  right  ascension  is  a  will 
culminate  at  a  hours,  sidereal  time. 

Therefore  the  sidereal  time  at  any 
meridian  is  equal  to  the  right  ascension 
of  that  meridian. 

In  the  figure  let  EE'  be  the  equator, 
Pthe  pole,  PM  the  meridian  of  any 
place,  PN  the  hour-circle  of  any  star 
5,  f  the  vernal  equinox.  Then  from  our  definitions, 

MPN  •=.  hour-angle  of  star  S  =  t ; 
NP°p  =  right  ascension  of  star  5  =  a  ; 

—  the  sidereal  time  at  the  meridian  PM  =  ®. 


*  If  the  latitude  of  the  place  of  the  observer  is  south,  obviously  these  con- 
ditions will  be  reversed. 


164  PRACTICAL  ASTRONOMY.  §  91. 

Theref  jre  @  =  a  +  /.       ......     (I97) 

Thus,  if  we  have  by  any  method  determined  the  hour- 
angle  of  a  star,  this  equation  gives  the  sidereal  time  ;  a,  the 
right  ascension,  being  taken  from  the  ephemeris,  or  from  a 
star  catalogue. 

The  interval  between  two  successive  transits  of  the  sun  over  the 
upper  branch  of  the  meridian  is  an  APPARENT  SOLAR  DAY. 

The  hour-angle  of  the  sun  at  any  meridian  is  the  APPARENT 
TIME  at  that  meridian. 

Owing  to  the  annual  revolution  of  the  earth,  the  sun's 
right  ascension  is  constantly  increasing  ;  therefore  it  follows 
that  the  solar  day  will  be  longer  than  the  sidereal  day. 
Thus  in  one  year  the  sun  moves  through  24  hours  of  right 
ascension.  In  one  year  there  are,  according  to  Bessel, 
365.24222  mean  solar  days;  therefore  in  one  day  the  sun's 

?4h 
right  ascension  increases  -^—  —  -  —  —  3m  $6*.$$$.    In  one  hour 


one  twenty-fourth  of  this  amount  =  98. 

These  figures  represent  the  mean  or  average  rate  of  change. 
The  actual  change,  however,  is  not  uniform,  and  in  conse- 
quence the  apparent  solar  days  are  not  of  equal  length.  This 
want  of  uniformity  results  from  two  causes,  which  will  now 
be  explained. 

First  Inequality  of  the  Solar  Day. 

91.  The  apparent  orbit,  of  the  sun  about  the  earth  is  an  el- 
lipse with  the  earth  in  one  of  the  foci.  Let  the  ellipse,  Fig.  12, 
represent  this  apparent  orbit.  When  the  sun  is  at  A  the 
right  ascension  is  increasing  more  rapidly  than  when  it  is  at 
A'  '•  therefore  in  the  first  case  it  will  have  a  larger  arc  to 
pass  over  between  two  successive  meridian  passages  than 
in  the  second.  This  inequality  alone  being  considered,  the 


92. 


INEQUALITY  OF  SOLAR  DAYS. 


I65 


length  of  the  solar  day  will  be  a  maximum  when  the  sun  is 

in  perigee,  and  a  minimum  when  it  is  in  apogee.     We  may 

imagine  a  fictitious  sun  to  move  in 

the  ecliptic  in  such  a  way  that  the  an- 

gular distances  AEP,  PEP,,  PEP,', 

etc.,  described  in  equal  times,  shall 

be  equal.     Let  both  start  together 

from  A  on  January  ist,  moving  in 

the  direction  of  the  arrow.     On  Jan- 

uary 2d  the  true  sun  will  be  in  ad- 

vance of  the  fictitious  sun,  and  will  FIG.  12. 

continue  so  until  June  3Oth,  when  they  will  be  together  at 

A  '.     Therefore  from  January  ist  to  June  3oth  the  fictitious 

sun,  having  the  smaller  right  ascension,  will  always  pass  the 

meridian  in  advance  of  the  true  sun.     From  A'  to  A  the 

fictitious  sun  will  be  in  advance  of  the  true  sun,  and  will  con- 

sequently pass  the  meridian  later,  until  they  both  reach  A, 

when  they  will  again  be  together,  January  ist. 

Second  Inequality  of  the  Solar  Day. 

92.  The  figure  represents  a  projection  of  the  sphere  on 
the  plane  of  the  equinoctial  colure.     P  is  the  north  pole,  Pr 

the  south  pole,  T  0=^  the  equa- 
tor, TSB^v3the  ecliptic.  Now 
the  fictitious  sun  before  con- 
sidered moves  in  the  ecliptic 
describing  the  equal  arcs  °pA, 
AB,  BCy  etc.,  in  equal  times. 
Let  the  hour-circles  PAP. 
PBP',  etc.,  be  drawn;  then 


the  distances 

intercepted   on  the  equator, 

will   not   be   equal,    but   the 

distance  T©  =  TO,  both  being 

quadrants. 


1 66  PRACTICAL   ASTRONOMY.  §  9-\ 

We  may  now  suppose  a  second  fictitious  sun  to  move  in 
the  equator  in  such  a  way  as  to  complete  the  circuit  of  the 
equator  in  the  same  time  that  the  first  completes  the  circuit 
of  the  ecliptic. 

Let  both  start  from  the  vernal  equinox  <?  together  on 
March  2Oth  ;  on  March  2ist  the  second  fictitious  sun  will  be 
in  advance  of  the  first,  and  will  continue  so  until  June  2Oth, 
when  they  will  both  have  completed  a  quadrant  and  will  be 
on  the  solstitial  colure  at  the  same  instant,  tthe  first  at  ®  and 
the  second  at  o.  Therefore  from  March  2ist  until  June 
2Oth  the  right  ascension  of  the  first  fictitious  sun  will  be  less 
than  that  of  the  second,  and  it  will  always  pass  the  meridian 
first. 

From  June  2Oth  to  September  22d  the  first  fictitious  sun 
will  be  in  advance  of  the  second,  at  which  time  they  will 
both  be  together  at  — .  From  September  22d  until  Decem- 
ber 2  ist  the  second  will  be  in  advance  of  the  first,  at  which 
time  they  will  both  again  be  on  the  solstitial  colure  at  the 
same  instant,  the  first  at  V3  and  the  second  at  o.  From  this 
until  March  2Oth  the  first  will  again  be  in  advance  of  the 
second,  when  finally  they  will  again  be  together  at  °P,  having 
completed  an  entire  revolution. 

As  the  second  fictitious  sun  describes  equal  arcs  of  the 
equator  in  equal  times,  it  follows  that  the  intervals  of  time 
between  each  two  successive  transits  over  the  same  branch 
of  the  meridian  will  be  equal. 

A  MEAN  SOLAR  DAY  is  the  interval  between  two  successive 

transits  of  the  second  fictitious  sun,  or  the  mean  sun  over  the 

upper  branch  of  the  meridian. 
THE  MEAN  SOLAR  TIME  at  any  meridian  is  the  hour-angle  of 

the  second  fictitious  sun  or  the  mean  sun  at  that  meridian. 
THE  EQUATION  OF  TIME  is  the  quantity  which  must  be  added 

algebraically  to  the  apparent  time  to  produce  the  mean  time. 


§  92-  EQUATION  OF    TIME.  1 67 

The  equation  of  time  is  given  in  the  Nautical  Almanac,  p. 
326  and  following,  for  Washington  apparent  noon  of  each 
day  in  the  year.  If  we  require  its  value  for  any  other  time, 
we  must  interpolate  between  the  values  there  given.  It  is 
the  algebraic  sum  of  the  two  inequalities  explained  above. 
From  the  foregoing  we  readily  see  that  the  equation  of  time 
will  be  zero  four  times  in  the  course  of  the  year ;  also  that 
there  will  be  two  maxima  and  two  minima  values. 

By  referring  to  the  ephemeris  for  1881,  we  find  the  value 
to  be  zero  on  April  I4th,  June  I3th,  August  3ist,  and  De- 
cember 23d.  The  maxima  values  -f-  I4m  28s  and  +  6m  15s 
occur  February  loth  and  July  25th  respectively  ;  the  mini- 
ma values  —  3m  51"  and  —  i6m  i8s  on  May  I4th  and  Novem- 
ber 2d. 

We  have  the  following  simple  precepts  : 

To  convert  a  given  instant  apparent  time  at  any  meridian  into 
the  corresponding  mean  time,  add  algebraically  to  the  apparent 
time  the  equation  of  time  taken  from  the  ephemeris. 

To  convert  the  mean  time  at  any  meridian  into  the  correspond- 
ing apparent  time,  subtract  the  value  of  the  equation  of  time 

taken  from  the  ephemeris. 

•       » 

Example  i.  1881,  July  4th,  5h  7™  i6s,  Bethlehem  apparent 
time  ;  find  the  corresponding  mean  time. 

Longitude  of  Bethlehem       -  6m  4OS.3 
Bethlehem  apparent  time    5h  7m  i68. 
Washington  apparent  time  5h  om  35s.  7  =  July  4.21 

From  the  Nautical  Almanac  (p.  329)  we  find 

Eq.  of  time  July  4  =  -f-  4m  iis.3o 
July  5  =  -\-  4m  2is.69 

Difference  ios.39 


168  PRACTICAL  ASTRONOMY.  §93. 

.21    X    IOS.39  =  2S-18 

Eq.  of  time  July  4       =         4™  i  is.3O 

July  4.21  =         41U  I3S.48 
Apparent  time  =  5h    7'"  i6s. 

Mean  time  =  5h  i  im  29*48 

Example  2.  1881,  November  i2th,  ioh  15^  7",   Bethlehem 
mean  time;  find  the  apparent  time. 
From  the  Nautical  Almanac  we  find 

Equation  of  time  —      -  15™  34s-7i 
Mean  time  =  ioh  15™    7s.oo 


Apparent  time        ioh  30™  41  \J\ 

Comparative  Length  of  the  Sidereal  and  Mean  Solar  Unit. 

93.  Owing  to  the  annual  revolution  of  the  earth  about  the 
sun,  the  number  of  sidereal  days  in  a  year  will  be  greater  by 
one  than  the  number  of  mean  solar  days.  According  to 
Bessel  the  year  contains 

365.24222  mean  solar  days;* 
366.24222  sidereal  days. 

Therefore 

366.24222    .  ,        .  , 
One  mean  solar  day  =  jg —      ^  sidereal  days 

=  1.00273791  sidereal  days; 

One  sidereal  day        =  ^j£~^  mean  S°lar  dayS 
=  0.99726957  mean  solar  days. 

*  These  values  given  for  1800  are  not  absolutely  constant;  the  length  of  the 
year  is  diminishing  at  the  rate  of  o§.595  in  100  years. 


§  93-  SIDEREAL  AND   MEAN  SOLAR    TIME.  169 

Let  /©  =  mean  solar  interval; 

/#  =  sidereal  interval; 
M  =  1.00273791. 

Then 

7*  =  lop  =  70  +  /©O  -  i)  ==  70  +  .002737917©;  ) 

4  7  7  (  M        r  r      I 

70  =    *      =  7*   -  7^1  ~  -£/=  7*  —  .002730434.  ) 

By  the  use  of  these  formulas  the  process  is  very  simple. 
It  is  rendered  still  more  so  by  the  use  of  tables  II  and  III 
of  the  appendix  to  the  Nautical  Almanac.  Table  II  gives 

the  quantity  (^  --  —  J7#,  with  the  argument  7^,  and  table  III 

gives  (/*  —  i)70,  with  the  argument  70. 

One  or  two  examples  will  illustrate  their  use. 

Example  i.  Given  the  mean  solar  interval  70  =  4h  40™  30". 
Find  the  corresponding  sidereal  interval. 

7©  =  4h  4om  30s.ooo 

Table  III  gives  for  4h  40™  -f-  45s-997 

Table  lit  gives  for      3os  +       .082 


=  4h 


Example  2.  Given  the  sidereal  interval  7^  =  4h  4im  i6s.o79 
Find  the  corresponding  mean  solar  interval. 


Table  II  gives  for  4h  4im  —  46S.O35 

Table  II  gives  for      i68.o79  «°44 

7©  —  4h  40m  30s.ooo 


17°  PRACTICAL  ASTRONOMY.  g  94. 

70  Convert  the  Mean  Solar  Time  at  any  Meridian  into  the  Cor- 
responding Sidereal  Time. 

94.  Referring  to  Fig  n  and  formula  (197),  we  see  that  if 
.S  represents  the  mean  sun,  then 

MPN  =  the  mean  time  =  T; 

=  the  right  ascension  of  the  mean  sun  =  «©. 


Then  we  have  ©  —  a®  -f-  T.     .     .    .  ,  .     .     .     (199) 

The  right  ascension  of  the  mean  sun,  a©,  is  given  in  the 
solar  ephemeris  of  the  Nautical  Almanac,  for  Washington 
mean  noon  of  each  day.  It  is  there  called  the  sidereal  time 
of  mean  noon,  which  it  is  readily  seen  is  the  right  ascension 
of  the  mean  sun  at  noon,  since  at  mean  noon  the  mean  sun 
is  on  the  meridian  when  itl  right  ascension  is  equal  to  the 
sidereal  time. 

If  L  =  the  longitude  of  the  meridian  from  which  T  is  reck. 
oned,  then  (T-\-  L)  =  the  time  past  Washington  mean  noon. 

Let  VQ  =  sidereal  time  of  mean  noon  at  Washington. 


Then  aQ  =  VQ  +  (T  +  L)(n  -  i), 

and  9  =  T  +  Vo  +  (T  +  L)(j*  -  i).     .     .     (200) 


The  last  term  may  be  taken  from  table  III  before  used,  or 
we  may  compute  it  by  the  method  given  in  Art.  90.  We  there 
found  the  hourly  change  in  right  ascension  of  the  mean  sun 
to  be  9S.8565.  If  we  express  (T  -\-  L)  in  hours,  we  have 


When  this  operation  has  frequently  to  be  performed  at 
any  meridian  other  than  Washington,  it  is  a  little  more  con- 
venient to  use  the  sidereal  time  of  mean  noon  at  the  merid- 
ian "itself. 

Let  V  =  the  sidereal  time  of  mean  noon  at  meridian  whose 


§94-  SIDEREAL  AND  MEAN  SOLAR    TIME.  I/I 

longitude  is  L.  Then  if  we  consider  L  as  reckoned  towards 
the  west,  the  Washington  time  of  mean  noon  at  the  given 
meridian  will  be  L,  and  we  shall  have 

V=  Fo  +  £  O  -  i), 
or          V  =  FO  +  9s.8s65Z;        L  being  expressed  in  hours. 

Formula  (200)  then  becomes 

&=  y+  T+  7\n  -  i).    .    .    .    .    (201) 

Example  i.  Longitude  of  Bethlehem  =—  6m4os.3  —  —  h.ni2; 

Mean  solar  time,  1881,  July  4th,  9h  oom  oos. 
Required  the  corresponding  sidereal  time. 
From  the  Nautical  Almanac,  p.  329,  we  find 

FO  =    6h  5im  228.6io 
—  -  .1112   x  9s.8s65,  or  from  table  III, 
N.  A.,  O  —  i)L  - 


V  '  —    6h  5im  2 
Mean  solar  time  T'=    9h  oom  oo'.ooo 

Table  III,  (JJL  -  i)T  +  im  28*708 


Sidereal  time  6)  =  i5h  52™  5os.222 

Example  2.   T  —  1881,  July  4th,  2  ih  7m3s.2,  Ann  Arbor  mean 
time.     Required  6). 

Longitude  of  Ann  Arbor  =  -|~26m  ^s.  i     =  h. 


VQ   =     6h  5lm  22s.6lO 

•4453  X  9s-8565>  or  table  III,  (p  -  i)Z  + 


V  '  =    6h  5im  2 

T=  2ih    7m    3s.2oo 

Table  III,  (/i  —  i)T  =     +  3rn  28s.  145 

Sidereal  time  0  =    4h  oim  5  8s.  344 


I/2  PRACTICAL   ASTRONOMY.  §  95. 

To  Convert  Sidereal  into  Mean  Solar  Time. 

95.  This  process,  the  converse  of  the  preceding,  may  be 
briefly  stated  as  follows: 

First.  Subtract  from  the  given  sidereal  time  the  sidereal 
time  of  mean  noon;  we  then  have  the  sidereal  interval  past 
noon,  viz.,  0  —  V. 

Second.  Convert  the  sidereal  interval  (0  —  V)  into  the 
corresponding  mean  time  interval,  by  subtracting  the  quan- 
tity (0  —  F)(i  -  -)  found  in  table  II,  N.  A. 

The  formula  is  as  follows: 


T=(Q—  V)  -  (0  --  F)(l  --)... 


(202) 


Example  i.  Given  1881,  July  4th,  i$h  52™  5os.222  Bethlehem 

sidereal  time. 
Required  the  corresponding  mean  solar  time. 

0  —    I5h  52m  50S.222 

4s  before,  V—    6h  5im  21^514 

0  —  v—    9"  Oim  28s.;o8 

Table  II,  (0  -  F)(i-  i)  im  28*708 

Mean  time  T  —    9h  oom  oos. 

Example  2.  Given  1881,  July  4th,  4h  im  58S.344  Ann  Arbor 

sidereal  time. 
Required  the  mean  solar  time. 

0  =    4h    im  58S.344 
As  before,  V  —    6h  5iln  26^.999 

0  —  V  =  2ih  iom    i8. 


Table  II,  (0  -  F)(i-  i)  3m  28s.  145 

Meantime  T=  2ih    7m  O38.2 


§95-  SIDEREAL  AND  MEAN   SOLAR    TIME.  173 

It  is  sometimes  necessary  to  cortvert  mean  solar  time  into 
sidereal,  or  vice  versa,  in  reducing-  old  observations  made 
before  the  publication  of  the  solar  ephemeris  in  the  form  now 
employed.  Bessel's  Tabula  Regiomontance  furnish  the  data 
necessary  for  solving  the  problem  for  any  date  between  1750 
and  1850.  The  method  of  using  these  tables  for  this  purpose 
is  fully  explained  in  Art.  362  of  this  work. 


CHAPTER  IV. 

ANGULAR  MEASUREMENTS.— THE  SEXTANT.— THE  CHRO- 
NOMETER AND  CLOCK. 

96.  The  circles  of  astronomical  instruments  are  graduated 
continuously  from  zero  to  360°.  With  ordinary  field-instru- 
ments the  smallest  division  is  commonly  10',  though  sometimes 
less.  The  large  circles  of  fixed  observatories  are  graduated 
much  finer.  Fractional  parts  of  a  division  are  read  by  means 
of  the  vernier,  or  reading  microscope. 

The  edge  of  the  circle  on  which  the  division  is  marked  is 
called  the  limb.  The  circle  or  arm  which  carries  the  index 
is  called  the  alidade. 

The  vernier,  also  called  the  nonius,  is  an  arc  carried  by  the 
alidade,  and  graduated  in  the  manner  described  below,  for 
measuring  fractional  parts  of  a  division. 

Let  AB  (Fig.  14)  be  a  portion  of  the  limb  of  a  circle.     Each 

division  is  supposed  to  be  one 
degree  of  the  circle.  The  arc 
CD,  carried  by  the  alidade  and 
graduated  as  shown,  forms  a 
vernier. 

In  this  case  there  are  ten  divisions  on  the  vernier,  cover- 
ing a  space  equal  to  nine  .divisions  of  the  limb.  Each  space 
on  the  vernier  is  therefore  shorter  by  -^  of  one  degree  (equals 
6')  than  a  space  on  the  limb.  In  the  figure  the  index  coin- 
cides with  the  zero-point  of  the  limb;  division  one  of  the  ver- 
nier falls  behind  division  one  of  the  limb,  6';  division  two  of 


§  96.  THE   VERNIER.  1/5 

the  vernier  falls  behind  division  two  of  the  limb,  2  X  6'  =  12', 
etc.,  etc. 

The  method  of  using  the  vernier  will  now  be  clear  by  re- 
ferring to   Fig.   15.      In    this  ^ 

case  the  index  falls  between     ^ c| 

42°  and  43°  on  the  limb.     The 


n 


reading  of  the  circle  is  there-  *r~^  44 

fore  42°  plus  a  fractional  part 

of  a  degree.     This  fraction  is  given  by  the  vernier  as  follows : 

Looking  along  the  scale  until  we  find  a  line  of  the  vernier 

which  coincides  with  a  line  of  the  limb,  we  find  this  to  be  the 

case  with  the  one  marked  4.     Therefore,  following  down  the 

vernier  scale  towards  the  zero-point,  it  is  evident  that 

Line  3  of  the  vernier  is       6'  to  the  right  of  45°  of  the  limb; 

Line  2  of  the  vernier  is  2x6'  =i2f  to  the  right  of  44°  of  the  limb; 
Line  i  of  the  vernier  is  3  X  6'=  1 8'  to  the  right  of  43°  of  the  limb; 
Line  o  of  the  vernier  is  4X  6^24'  to  the  right  of  42°  of  the  limb. 

The  reading  is  therefore  42°  24'  or  42°.4,  the  number  on  the 
vernier  where  the  line  of  the  latter  coincides  with  a  line  of 
the  limb,  giving  the  tenths  of  a  degree  at  once. 
In  general  let 

d  =  the  value  of  one  division  of  the  limb; 
d'  =  the  value  of  one  division  of  the  vernier ; 
n  =  the  number  of  divisions  of  the  vernier  corresponding  to 
n  —  i  of  the  limb. 

Then  (n  —  i)d  =  nd' , 

and  d—d'  —  -d.  ......     (203) 

d  —  d'  is  the  least  reading  of  the  vernier.  We  have  therefore 
the  following  very  simple  rule : 


PRACTICAL  ASTRONOMY.  §  9/. 

To  find  the  least  reading  of  a  vernier :  Divide  the  length  of  one 
division  of  the  limb  by  the  number  of  spaces  of  the  vernier. 

For  example,  suppose  the  limb  graduated  to  10',  and  the 
number  of  divisions  of  the  vernier-scale  to  be  60.  Then  the 
least  reading  of  the  vernier  will  be 

10'      600" 


60  "  60 

This  is  a  very  common  arrangement. 

In  the  vernier  just  described  n  divisions  of  the  vernier 
were  equal  to  n  —  I  of  the  limb.  Verniers  are  sometimes 
made  in  which  n  divisions  are  equal  to  n  +  I  of  the  limb. 

Then  (n  +  i)d  —  nd     and     d  —  d  —  -d,      as  before. 

It  is  to  be  observed  that  in  this  case  the  reading  of  the  ver- 
nier proceeds  in  a  direction  opposite  to  that  of  the  limb. 

Many  different  forms  of  division  and  arrangement  are 
found  in  verniers,  but  they  all  follow  the  same  general  princi- 
ple, a  practical  familiarity  with  which  makes  the  reading  of 
any  form  of  vernier  very  simple. 

The  Reading  Microscope. 

97.  Instead  of  the  vernier,  in  very  fine  instruments  the 
alidade  carries  a  microscope  the  optical  axis  of  which  is  per- 
pendicular to  the  plane  of  the  circle.  This  is  a  compound 
microscope  with  a  positive  eye-piece.  In  the  common  focus 
of  the  object-lens  and  eye-piece  are  the  micrometer-threads 
for  reading  the  circle.  The  micrometer  (Fig.  160)  consists  of 
a  frame  of  brass,  across  which  are  stretched  two  spider-lines. 
Sometimes  these  lines  make  an  acute  angle  with  each  other, 
as  shown  in  the  figure ;  sometimes  they  are  made  parallel  and 
quite  close  together.  The  plane  of  the  frame  is  parallel  to 


§97- 


THE  READING  MICROSCOPE. 


the  plane  of  the  circle  MNt  and  it  is  moved  parallel  to  a  tan- 
gent to  the  circle  by  the  screw  G.  Attached  to  the  screw  and 
revolving  with  it  is  the  cylinder  FE,  graduated,  as  shown  in 
the  figure,  for  recording  the  fractional  parts  of  a  revolution  of 
the  screw.  The  cylinder  is 'generally  graduated  into  either 
60  or  loo  parts.  Suppose  now  the  distance  between  two 
ui visions  of  the  circle  to  be  5',  and  that  five  revolutions  of 
tae  screw  are  just  sufficient  to  move  the  cross-threads  over 
this  distance :  then  evidently  one  revolution  moves  the  threads 
over  i'.  If  the  head  is  divided  into  60  parts,  then  each  divi- 
sion of  the  head  corresponds  to  a  motion 
of  the  cross-threads  over  i".  By  making 
the  screw  sufficiently  fine  and  increasing 
the  number  of  divisions  of  the  head,  at 
the  same  time  increasing  the  power  of 


FIG.  i6a.---THE  MICROMETER. 


FIG.  16. — THE  READING  MICROSCOPE. 


the  microscope,  this  division  of  space  may  be  carried  to  an 
almost  unlimited  extent.  For  the  purpose  under  considera- 
tion, however,  we  should  soon  reach  a  limit  beyond  which 
nothing  would  be  gained  by  increasing  the  delicacy  of  the 
microscope. 

For  reading  the  entire  number  of  revolutions  of  the  screw 
there  is  sometimes  a  scale  attached  to  the  outside  of  the  box  in 
which  the  slide  moves.  More  frequently  the  scale  is  inside 
the  box,  placed  at  one  side  of  the  field  of  view.  When  so 
placed  it  consists  of  a  strip  of  metal  in  the  edge  of  which 


178  PRACTICAL  ASTRONOMY.  §  98. 

notches  are  cut ;  the  distance  between  two  consecutive  notches 
being  equal  to  one  revolution  of  the  screw.  Every  fifth  notch 
is  made  deeper  than  the  others  for  facility  in  counting. 

Suppose  now  the  cross-threads  to  stand  opposite  the  centre 
notch  (which  is  generally  distinguished  in  some  manner),  and 
the  zero  point  of  the  head  to  be  exactly  at  the  index-mark. 
The  point  in  the  field  now  occupied  by  the  cross-threads  is 
the  fixed  point  to  which  all  angular  measurements  are  re- 
ferred ;  it  corresponds  exactly  to  the  zero-point  of  the  ver- 
nier. Suppose,  further,  the  zero-point  of  the  circle  to  be 
exactly  under  the  intersection  of  the  threads.  Now  let  the 
instrument  be  revolved  on  its  axis  through  any  angle :  the 
number  of  divisions  of  the  circle  which  pass  by  this  point 
of  reference  will  then  be  the  measure  of  the  angle. 

For  the  purpose  of  fixing  the  idea,  let  the  arrangement  be 
that  described  above,  viz.,  the  circle  graduated  to  5',  and  the 
micrometer  reading  to  single  seconds.  If  now  the  revolu- 
tion of  the  instrument  has  brought  the  scale  into  the  position 
shown  in  Fig.  17,  we  see  from  the  position  of  the  threads 
that  the  entire  angle  passed  over  is  between  45°  15'  and 
45°  20'.  By  means  of  the  screw  let  the  cross-threads  be 
moved  so  as  to  coincide  with  division  15'.  Then  the  entire 

' number  of  revolutions  of  the  screw  will 

give  the  number  of  minutes  to  be  added 
to  45°  15',  and  the  fractional  part  of  a 
revolution  given  by  the  head  will  be 
expressed  in  seconds.  Thus  if  the  whole 

number  of  revolutions  were  two,  and  the  reading  of  the  head 
53,  the  angle  would  be  45°  if  53".  In  making  the  bisection, 
the  screw  should  always  be  turned  in  the  same  direction,  to 
guard  against  the  effect  of  slip  or  lost  motion  in  the  screw. 
If  the  thread  is  to  be  moved  in  a  negative  direction  it  should 
be  moved  back  beyond  the  line,  and  the  final  bisection  made 
by  bringing  it  up  from  the  other  side. 

98.  When  everything  is  in  perfect  order  a  whole  number 


I 

§98.  THE   READING  MICROSCOPE.  179 

of  revolutions  of  the  screw  is  exactly  equal  to  the  distance 
between  two  consecutive  lines  on  the  circle.  This  is  pro- 
vided for  by  an  arrangement  for  changing  the  focal  length 
of  the  microscope,  and  for  moving  the  object-lens  nearer  to 
or  farther  from  the  plane  of  the  circle.  This  adjustment  is 
subject  to  small  disturbances,  on  account  of  changes  of 
temperature  and  other  causes.  The  error  caused  by  an  im- 
perfect adjustment  is  called  the  error  of  runs.  The  correc- 
tion for  runs  is  found  by  reading  the  microscope  on  two  con- 
secutive divisions  of  the  circle.  If  this  does  not  correspond 
to  the  exact  number  of  revolutions  of  the  screw,  the  excess 
or  deficiency  is  to  be  distributed  in  the  proper  proportion 
to  measurements  made  with  the  screw. 

For  determining  the  correction  a  number  of  readings 
should  be  made  in  different  parts  of  the  circle  in  order  to 
eliminate  from  the  result  the  accidental  errors  of  graduation. 
Some  observers  in  certain  kinds  of  work  always  read  the 
micrometer  on  both  divisions  of  the  limb  between  which  the 
zero-point  falls.  For  example,  in  Fig.  17  the  micrometer- 
thread  would  be  set  on  both  division  15'  and  20',  thus  eliminat- 
ing from  the  resulting  reading  the  effect  of  runs,  and  to  some 
extent  the  accidental  errors  of  graduation  and  of  bisection. 

For  insuring  greater  accuracy  two  or  more  microscopes 
or  verniers  are  used.  When  there  are  two  they  are  placed 
opposite  each  other,  or  180°  apart.  When  there  are  three 
or  more  they  are  placed  at  uniform  distances  around  the 
circle.  If  the  probable  error  of  the  reading  of  one  micro- 

i/> 
scope  be  \" ,  that  of  the  mean  of  two  will  be*— -  —  ".71 ; 

V2 
l" 

that  of  four  will  be  — ~  =  ".5. 

V4 

The  principal  value  of  two  or  more  microscopes,  however, 
is  for  eliminating  the  error  of  eccentricity. 

*  See  Introduction,  Art.  14,  Eq.  (25). 


I8o 


PRACTICAL   ASTRONOMY. 


§99- 


Eccentricity  of  Graduated  Circles. 

99.  The  centre  of  the  alidade  seldom  coincides  exactly 
with  the  centre  of  the  graduated  circle.  This  deviation 

from  exact  coincidence  is  called  ec- 
centricity. 

In  order  to  understand  the  effect 
of  eccentricity,  let 

C    be  the  centre  of  the  circle  ; 
C  ',  the  centre  of  the  alidade  ; 
O,  the  zero-point  of  the  limb  ; 
a,   the  point  on  the  limb  where  it  is 
intersected  by  a  line  joining  C 
and  C'  ; 

C'n,  the  direction  of  the  line  drawn  from  the  centre  of  the 
alidade  to  the  zero-point  of  the  vernier  when  the 
telescope  is  directed  to  any  object. 

The  true  position  of  the  object  is  given  by  the  direction 
of  the  line  C'n,  while  the  reading  of  the  circle  gives  the 
direction  Cn,  differing  from  the  former  by  the  small  angle 
n'Cn  =  CnC'. 


is. 


Let  now 


CnC  =  p  ; 

CC  =  e ; 

Cn  =  r ; 

Cn  =  r'; 


Angle  OCn  =  n\ 
OCa  =  a. 

Then  CCn  =  n  —  a. 


From  the  triangle  CCn  we  have 

rr  sin /  =  e  sin  (n  —  a); 

r'  cos/  =  r  —  e  cos  (n  —  a); 


from  which 


tan/  = 


-  sin  («  —  a) 


.     .     (204) 


§  IOO.          ECCENTRICITY  OF  GRADUATED    CIRCLES.  l8l 

The  angle  /  will  always  be  small,  and  the  denominator  of 
(204)  differs  but  little  from  unity.  We  may  therefore  write, 
without  appreciable  error, 

/  =  -  sin  (n  —  a)  ......     (205) 

IOO.  It  is  more  elegant  to  expand  the  above  expression  into  a  series  in  terms 
of  ascending  powers  of  -.     Equation  (204)  is  of  the  form 

sin/  _        a  sin  x 
cos/        i  —  a  cos  x  ' 

from  which  we  readily  find 

sin/  =  a  sin  (/  -|-  x)  .........     (206) 

Now  add  sin  (/  -f-  x)  to  both  members  of  (206)  ;  then  subtract  sin  (/  -|-  x)  from 
both  members  ;  finally,  divide  the  first  expression  by  the  second: 

sin  /  -f-  sin  (/  -f-  x)  _  (a  -\-  i)  sin  (p  -f  x)  t 
sin  /  —  sin  (/  -f-  x)  ~  (a  —  i)  sin  (/  -j-  x)  ' 

from  which  tan  (/  +  \x)  =  —     —  tan  \x  ........     (207) 

Applying  to  this  the  process  of  development  made  use  of  in  Art.  74,  Eq.  (137), 
we  find 

/  =  a  sin  x  -f-  $a?  sin  2x  -f-  £a3  sin  $x,  etc. 

Writing  for  a  and  x  their  values  and  dividing  by  sin  i",  in  order  to  express/  in 
seconds  of  arc,  we  find 


sn  M  ~ 


-V'  sin  2(B  -  a)  +         T'  sin  3("-tt)'  (208) 


The  first  term  is  identical  with  (205),  and  will  always  give  the  necessary  accu- 
racy without  using  the  following  terms. 

101.  Besides  the  eccentricity  above  considered  there  is  a 
similar  effect  due  to  the  play  of  the  axis  of  the  instrument  in 


1 82  PRACTICAL  ASTRONOMY.  §  IOI. 

its  socket.     This  is  not  a  determinate  quantity  like  that  we 
have  been  considering,  but  when  two  verniers  or  microscopes 
1 80°  apart  are  used,  the  effect  of  both  will  be  eliminated,  as    , 
appears  from  the  following : 

Let      ri  and  n"  be  the  readings  of  the  two  microscopes  ; 
#,  the  true  value  of  the  angle. 

Then  from  the  first  microscope 

n  —  n'  +  e"  sin  (ri  —  a). 
Similarly,  n  =  n"  +  e"  sin  (n"  -  a). 

In  which  e"  has  been  written  for 


r  sin  i"' 


Now  n"  differs  very  little  from  180°  +  ri  ,  so  that  no  appre- 
ciable error  will  be  introduced  by  writing  the  second  of  the 
above  equations 


n  = 


e"  sin  [180°  +  (ri  —  a]~\  =  n"  —  e"  sin  (ri  —  a). 


Therefore  n  =  \(ri  -f-  n"),  from  which  the  correction  for 
eccentricity  is  eliminated.  In  a  similar  manner  it  may  be 
shown  that  the  mean  of  three  microscopes  will  be  free  from 
the  effect  of  eccentricity.  In  case  of  four,  as  the  mean  of 
each  pair  180°  apart  is  free  from  this  error,  it  follows  that 
the  mean  of  the  four  will  be. 

The  constants  e"  and  a  may  be  determined  very  readily  by 
taking  readings  in  different  parts  of  the  circle  ;  but  with  a 
complete  circle  they  will  not  be  required.  It  is  only  in 
the  case  of  the  sextant,  where  we  have  a  limited  arc  of  the 
circle  read  by  a  single  vernier,  that  this  becomes  a  matter  of 
importance.  The  application  to  this  case  will  be  considered 
in  the  proper  place. 


102. 


THE   SEXTANT. 

The  Sextant. 


102.  In  the  determination  of  time  and  latitude  when  ex- 
treme accuracy  is  not  required,  the  sextant  is  one  of  the  most 
convenient  and  useful  of  astronomical  instruments.  It  is 
light  and  easy  of  transportation  ;  in  observing  it  is  simply 
held  in  the  hand,  and  consequently  entails  no  loss  of  time  in 


FIG.  19.— THE  SEXTANT. 

mounting  and  adjusting ;  it  is  therefore  especially  adapted  to 
the  requirements  of  navigation  and  exploration.  For  use  on 
land  the  sextant  is  sometimes  mounted  on  a  tripod,  which 
adds  something  to  its  accuracy.  When  the  instrument"  is 
used  by  a  skilful  observer,  however,  the  advantage  is  not 
great.  In  most  cases  where  such  an  arrangement  could  be 
made  use  of  the  sextant  will  not  be  employed  at  all,  but  will 
give  place  to  an  instrument  of  greater  precision. 


J84  .        PRACTICAL  ASTRONOMY.  §  103. 

The  principal  features  of  the  sextant  may  be  seen  from 
Fig.  19.  The  graduated  arc  is  about  60°  in  extent,  hence  the 
name,  sextant.  This  arc  of  60°  is  divided  into  120  parts, 
called  degrees  for  reasons  which  will  soon  appear.  The  arc 
commonly  reads  directly  to  ior,  and  by  means  of  the  vernier 
to  10".  A  mirror,  C,  called  the  index-glass,  is  attached  to  the 
arm  carrying  the  vernier,  and  revolves  with  it  about  a  pivot 
at  the  centre.  A  second  mirror,  Ny  is  attached  to  the  frame  of 
the  instrument,  and  is  called  the  horizon-glass.  Only  half  of 
this  glass  is  silvered,  viz.,  that  next  the  plane  of  the  instru- 
ment— an  arrangement  which  makes  it  possible  to  see  an  ob- 
ject directly  through  the  unsilvered  part  by  means  of  the 
telescope,  and  at  the  same  time  the  image  of  the  same  object, 
or  of  a  second  one,  reflected  from  the  silvered  part  of  the 
mirror.  In  order  to  make  these  images  equally  distinct  an 
adjusting-screw  is  provided  (not  shown  in  the  figure),  by 
which  the  telescope  can  be  moved  nearer  to  the  plane  of  the 
instrument  or  farther  from  it.  Attached  to  the  frame  are  sev- 
eral colored  glasses,  E  and  F,  which  may  be  brought  into  a 
position  to  protect  the  eye  when  observing  the  sun.  These 
are  sometimes  attached  to  an  axis  so  that  they  can  be  at  once 
reversed,  the  object  being  to  eliminate  any  error  due  to  want 
of  parallelism  of  the  surfaces  by  taking  half  of  a  series  of 
measurements  in  each  position.  There  is  also  a  revolving 
disk  attached  to  the  eye-piece  of  some  instruments  containing 
a  number  of  colored  glasses  of  different  shades.  Other  minor 
features  can  best  be  learned  by  the  inspection  of  the  instru- 
ment itself. 

103.  The  principle  which  lies  at  the  foundation  of  the  sex- 
tant and  instruments  of  like  character  is  the  following :  If  a 
ray  of  light  suffers  two  successive  reflections  in  the  same 
plane  by  two  plane  mirrors,  then  the  angle  between  the  first 
and  last  direction  of  the  ray  is  double  the  angle  of  the  mir- 
rors. In  Fig.  20  let  M  and  m  be  the  two  mirrors  supposed 


103- 


THE   SEXTANT. 


185 


perpendicular  to  the  plane  of  the  paper ;  let  AM  be  the  first 
direction  of  a  ray  of  light  falling  on  the  mirror  M\  it  will  be 
reflected  in  the  direction  Mm,  and  finally  from  m  in  the  direc- 
tion mE.  Draw  MB  parallel  to  mE,  MP  perpendicular  to  M, 
Mp  perpendicular  to  m.  The  angle  between  the  first  and 


FIG.  20. 

last  direction  of  the  ray  is  equal  to  the  angle  AMB.  The 
angle  between  the  mirrors  is  equal  to  PMp.  We  have  now 
to  show  that  A  MB  —  2PMp. 

Consider  first  the  mirror  m.     The  incident  ray  Mm  makes 
with  the  normal  the  angle 

Mmpf  =  mMp  =  pMB  =  pMP  +  PMB.     .    .    (a) 
Consider  now  M.     The  angle 

mMP  =  PMA  =  AMB  +  PMB (V) 


1 86  PRACTICAL  ASTRONOMY.  §  1O4. 

Subtracting  (a)  from  (£), 

mMP  —  mMp  —  AMB  -  pMP, 
from  which  2pMP  =  ^ MB.  Q.  E.  D. 

If  now  the  angle  between  two  objects  is  to  be  measured, 
the  instrument  is  held  so  that  the  plane  of  the  graduated  arc 
passes  through  both.  The  telescope  is  then  directed  to  one 
of  the  objects,  which  is  seen  through  the  unsilvered  part  of 
the  horizon-glass,  and  the  index-arm  is  revolved  until  the  re- 
flected image  of  the  second  object  is  brought  in  contact  with 
the  direct  image  of  the  first.  The  reading  of  the  limb  will 
then  be  the  required  angle  ;  the  graduation  before  explained, 
viz.,  each  degree  being  divided  into  two,  gives  the  angle 
between  the  objects,  which  is  twice  that  of  the  mirrors. 

104.  In  the  prismatic  sextant  of  Pistor  &  Martins  (Fig.  21) 
the  horizon-glass  is  replaced  by  a  totally  reflecting  prism. 
The  arrangement  has  this  advantage,  viz.,  that  by  its  use 
angles  of  all  sizes  from  o°  to  180°,  and  even  larger,  can  be 
measured,  while  the  common  form  of  sextant  is  not  adapted 
to  the  measurement  of  angles  much  greater  than  120°. 

In  using  the  instrument  the  prism  B  interferes  with  the 
rays  of  light  which  should  reach  the  index-glass,  A,  when 
the  angle  is  about  140° ;  but  angles  of  this  magnitude  may 
be  measured  by  turning  the  instrument  over  and  holding  it 
in  the  reverse  position.  If,  for  instance,  the  double  altitude 
of  the  sun  is  being  measured,  the  instrument  will  ordinarily 
be  held  in  the  right  hand,  with  the  arc  below  and  the  tele- 
scope above.  If,  however,  the  double  altitude  is  about  140°, 
the  instrument  must  be  held  in  the  left  hand,  with  the  tele- 
scope below  and  the  arc  above.  In  case  the  head  of  the  ob- 
server interferes,  as  will  be  the  case  when  the  angle  is  near 
1 80°,  the  difficulty  is  overcome  by  means  of  the  prism  E 


§  105.      PRISMATIC  SEXTANT.— REFLECTING   CIRCLE.  l8/ 

placed  back  of  the  eye-piece  so  as  to  reflect  the  rays  of  light 
coming  through  the  telescope  in  a  direction  at  right  angles 
to  its  axis. 

105.  The  arc  of  the  sextant  may  be  extended  to  an  entire 
circumference,  and  the  index-arm  produced  so  as  to  carry  a 


FIG.  2i.—  THE  PRISMATIC  SEXTANT. 


vernier  at  each  extremity.  The  instrument  then  becomes 
the  simple  reflecting  circle.  As  previously  shown,  this  arrange- 
ment possesses  the  advantage  of  eliminating  the  eccentricity, 
and  to  some  extent  the  errors  of  graduation.  This  instru- 
ment is  used  precisely  like  the  sextant. 


1 88  PRACTICAL  ASTRONOMY.  §  IO;. 

Other  forms  of  reflecting  circles  have  been  made  possess- 
ing advantages  in  certain  directions,  but  they  do  not  seem 
to  have  met  with  great  favor,  although  they  are  theoretically 
much  more  perfect  instruments  than  the  sextant ;  practically, 
however,  this  superiority  is  not  so  great.  This  is  no  doubt 
due  in  part  to  the  fact  that,  except  in  the  hands  of  an  obser- 
ver of  more  than  usual  skill,  the  errors  of  observation  are  so 
great  as  practically  to  neutralize  their  greater  theoretical 
advantages. 

Adjustments  of  the  Sextant. 

1 06.  First  Adjustment.     THE  INDEX-GLASS.     The  plane  of 
tlie  reflecting  surface  must  be  perpendicular  to  the  plane  of  the 
sextant. 

To  ascertain  whether  this  is  the  case,  place  the  index  near 
the  middle  of  the  arc,  then  look  into  the  glass  so  as  to  see 
the  image  of  the  arc  reflected.  If  the  adjustment  is  perfect, 
the  arc  seen  directly  will  be  continuous  with  its  reflected 
image. 

This  adjustment  is  attended  to  by  the  maker  and  is  not 
liable  to  derangement ;  for  this  reason  no  provision  is  com- 
monly made  for  correcting  a  want  of  perpendicularity.  It 
may  be  corrected  when  necessary  by  removing  the  glass 
from  its  frame  and  filing  down  one  of  the  points  against 
which  it  rests,  or  by  loosening  the  screws  holding  the  frame 
to  the  index-arm  and  inserting  a  piece  of  paper  or  other  thin 
substance  under  one  side. 

107.  Second  Adjustment.    THE  HORIZON-GLASS.     The  plane 
of  this  mirror  must  also  be  perpendicular  to  the  plane  of  the 
sextant. 

The  index-glass  must  first  be  in  adjustment;  if  then  it  is 
possible  to  place  it  in  a  position  parallel  to  the  horizon-glass 
by  moving  the  index-arm,  then  the  latter  will  also  be  per- 
pendicular to  the  plane  of  the  sextant.  To  test  this  adjust- 


§  108.  ADJUSTMENT  OF   THE   SEXTANT.         '  189 

ment  proceed  as  follows  :  Bring  the  index  near  the  zero- 
point  and  direct  the  telescope  to  a  well-defined  point — a  star 
is  best.  If  then  the  index-arm  be  moved  slightly  one  way 
and  then  the  other — the  plane  of  the  instrument  being  verti- 
cal— the  reflected  image  of  the  object  will  move  up  and  down 
through  the  field.  If  the  adjustment  of  the  two  glasses  is 
perfect,  the  two  images  may  be  made  to  coincide  exactly, 
otherwise  the  reflected  image,  instead  of  passing  over 
the  direct,  will  pass  to  one  side  or  the  other  of  it.  Two 
small  capstan-headed  screws  are  provided  for  making  this 
adjustment  when  necessary.  A  pair  of  adjusting-screws  is 
also  provided  for  correcting  the  position  of  the  glass  in  the 
opposite  direction,  viz.,  to  make  it  parallel  to  the  index-glass 
when  the  vernier  is  at  zero.  If  the  direct  and  reflected 
image  of  the  star  are  brought  into  exact  coincidence  by 
means  of  the  tangent-screw,  the  reading  of  the  vernier,  if 
not  zero,  is  called  the  index  error.  The  screws  just  men- 
tioned are  for  correcting  this  error.  It  will  be  found  better 
in  practice  not  to  attempt  this  adjustment,  but  to  determine 
the  error  and  apply  the  necessary  correction  to  the  angles 
measured,  as  will  be  explained  hereafter. 

1 08.  Third  Adjustment.  The  axis  of  the  telescope  must  be 
parallel  to  the  plane  of  the  instrument. 

Two  parallel  threads  are  placed  in  the  eye-piece  to  mark 
approximately  the  middle  of  the  field:  they  should  be  made 
parallel  to  the  plane  of  the  instrument  by  revolving  the  eye- 
piece. The  axis  of  the  telescope  will  now  be  the  line  drawn 
through  the  optical  centre  of  the  object-glass  and  a  point 
midway  between  these  lines.  To  determine  whether  this 
line  is  parallel  to  the  plane  .of  the  instrument,  select  two 
well-defined  objects  100°  or  more  apart,  and  bring  the  re- 
flected image  of  one  in  contact  with  the  direct  image  of  the 
other,  making  the  contact  on  one  of  the  threads ;  then  move 
the  instrument  so  as  to  bring  the  images  on  the  other  thread. 


1 90  PRACTICAL  ASTRONOMY.  §  1 09. 

If  the  contact  still  remains  perfect,  the  line  is  in  adjustment ; 
if  any  correction  is  required,  there  will  be  found  a  pair  of 
screws  for  the  purpose  on  opposite  sides  of  the  ring  which 
holds  the  telescope. 

The  above  test  will  be  found  difficult  to  apply,  especially 
if  the  observer  has  not  a  considerable  amount  of  experience 
in  the  use  of  the  instrument.  One  less  difficult  is  the  follow- 
ing :  Place  the  instrument  face  upward  on  a  table,  then  lay 
on  the  arc  two  strips  of  metal  or  wood,  the  width  of  which 
must  be  the  same  and  equal  to  the  distance  of  the  axis  of  the 
telescope  from  the  plane  of  the  instrument.  Now  sight 
across  the  upper  edges  of  these  strips,  and  have  an  assistant 
mark  with  a  pencil  on  the  wall  of  the  room  (which  should  be 
15  or  20  feet  distant)  the  place  where  the  sight-line  inter- 
sects it;  then,  without  disturbing  anything,  look  through  the 
telescope,  which  has  been  previously  directed  to  this  part  of 
the  wall  and  properly  focused,  and  see  whether  this  mark 
is  found  in  the  middle  of  the  field  ;  if  so,  then  the  adjustment 
is  satisfactory. 

Method  of  Observing  with  the  Sextant. 

109.  To  Measure  the  Distance  between  Two  Stars.  Direct 
the  telescope  to  one  of  the  stars,  then  revolve  the  instrument 
about  the  axis  of  the  telescope  until  its  plane  passes  through 
the  other  (taking  care  to  have  the  index-glass  on  the  right 
side),  then  move  the  index-arm  until  the  image  of  the  second 
star  is  brought  into  the  field,  clamp  the  instrument  and  bring 
the  two  images  into  perfect  contact  by  means  of  the  tangent- 
screw.  The  reading  of  the  vernier  corrected  for  index  error 
will  be  the  required  distance.  Unless  the  two  stars  are  quite 
near  each  other  it  will  be  expedient  to  compute  the  distance 
approximately  before  attempting  the  observation.  The  in- 
dex  may  then  be  set  at  the  approximate  distance,  which  will 


§  1 10.         METHOD   OF  OBSERVING   WITH  SEXTANT.  IQI 

greatly  facilitate  finding-  the  two  images.  A  common  obser- 
vation of  this  character  is  that  of  observing  the  distance  of 
the  moon  from  the  sun  or  a  star  for  determining  longitude. 
In  the  Nautical  Almanac  will  be  found  given  for  every  day 
throughout  the  year  the  distance  of  the  moon  from  the  sun, 
and  certain  stars  and  planets,  which  may  be  used  for  this 
purpose.  The  index  may  at  once  be  set  at  the  approximate 
angle  without  any  preliminary  computation.  If  the  distance 
of  the  moon  from  a  star  is  measured,  the  image  of  the  star  is 
brought  into  contact  with  the  bright  limb  of  the  moon,  the 
contact  being  made  at  the  point  where  the  great  circle  join- 
ing the  star  with  the  centre  of  the  moon  intersects  the  limb. 
To  ascertain  this  point  the  instrument  must  be  revolved 
through  a  small  arc  back  and  forth  about  the  axis  of  the 
telescope  (supposed  to  be  directed  to  the  star);  the  image  of 
the  moon's  limb  will  then  pass  back  and  forth  across  the 
field,  and  should  appear  to  pass  exactly  through  the  centre 
of  the  star's  image,  which  will  in  general  not  be  reduced  to 
a  simple  point  by  the  feeble  telescope  of  the  sextant. 

This  distance  is  to  be  corrected  for  the  moon's  semidiam- 
eter  in  order  to  give  the  distance  between  the  star  and  the 
centre  of  the  moon. 

In  measuring  the  distance  between  the  moon  and  sun,  the 
bright  limb  of  the  moon  is  brought  in  contact  with  the  near- 
est limb  of  the  sun.  The  measured  distance  must  then  be 
corrected  for  the  semidiameters  of  both  moon  and  sun. 

no.  Measurement  of  Altitudes.  At  sea  altitudes  are  meas- 
ured by  bringing  the  reflected  image  of  the  body  in  contact 
with  the  line  of  the  horizon  as  seen  directly  through  the 
telescope.  In  order  that  the  result  may  be  correct  the 
plane  of  the  instrument  must  be  held  exactly  vertical.  To 
accomplish  this  the  instrument  is  revolved  or  vibrated 
slightly  about  the  axis  of  the  telescope,  at  the  same  time 
moving  it  so  as  to  keep  the  image  in  the  centre  of  the  field. 


1 92  PRACTICAL  ASTRONOMY.  g  U  £ 

The  image  will  appear  to  describe  an  arc  of  a  circle,  the 
lowest  point  of  which  must  be  made  tangent  to  the  horizon 
by  moving  the  index-arm.  If  the  sun  is  observed,  the  lower 
limb  must  be  made  tangent  to  the  horizon.  As  the  altitude 
of  the  sun's  centre  is  required,  the  reading  of  the  vernier 
must  be  corrected  for  index  error,  refraction,  parallax,  and 
semidiameter.  If  a  star  is  observed,  there  will  be  no  correc- 
tion for  semidiameter  or  parallax. 

in.  For  observing  altitudes  on  land  the  artificial  horizon 
must  be  used.  This  is  a  shallow  basin,  about  3  inches  by  5, 
for  holding  mercury.  It  is  provided  with  a  roof  formed  of 
two  pieces  of  plate  glass  set  at  right  angles  to  each  other  in 
a  metal  frame,  for  protecting  the  mercury  from  agitation  by 
the  wind.  The  surface  of  the  mercury  forms  a  mirror  from 
which  the  image  of  the  sun  or  star  is  reflected  ;  and  as  it  is 
perfectly  horizontal  the  reflected  image  will  appear  at  an 
angular  distance  below  the  horizon  equal  to  the  altitude  of 
the  body  itself  above  the  horizon.  If  now  the  image  of  a 
star  reflected  from  the  mirrors  of  the  sextant  is  brought  into 
contact  with  the  image  reflected  from  the  mercury,  the  angle 
which  will  be  measured  is  evidently  twice  the  altitude  of  the 
star. 

The  opposite  sides  of  the  glass  plates  forming  the  roof  to 
the  horizon  should  be  exactly  parallel,  otherwise  the  pris- 
matic form  introduces  an  error  into  the  measured  angle.  It 
is  possible  to  derive  a  formula  for  the  correction  necessary 
to  free  an  observation  from  this  source  of  error,  but  it  will 
be  better  in  practice  to  observe  half  of  a  series  of  altitudes 
with  one  side  of  the  roof  next  the  observer  and  then  reverse 
it,  taking  the  remaining  half  in  the  opposite  position. 

The  mercury  must  be  freed  from  the  particles  of  dust  and 
impurities  which  will  generally  be  found  floating  on  its  sur- 
face. It  may  be  strained  through  a  piece  of  chamois-skin 
or  through  a  funnel  of  paper  brought  down  to  a  fine  point 


§113-         METHOD   OF  OBSERVING   WITH  SEXTANT.  1 93 

at  the  end.  Another  method  is  to  add  a  small  amount  of  tin- 
foil to  the  mercury,  when  the  amalgam  which  will  be  formed 
will  rise  to  the  top  and  may  be  drawn  to  one  side  with  a 
card,  leaving  the  surface  entirely  free  from  specks  of  any 
kind. 

112.  In  measuring  altitudes  for  any  purpose,  a  number  of  measures  should 
be  made  in  quick  succession  and  the  mean  taken.      In  this  way  the  accidental 
errors  of  contact  and  reading  will  be  greatly  diminished.     Thus,  in  taking  the 
altitude  of  the  sun  for  determining  tne  time,  a  series  of  not  less  than  three  alti- 
tudes should  be  measured  on  each  limb.     Suppose  the  observations  made  when 
the  sun  is  east  of  the  meridian,  and  the  altitudes  therefore  to  be  increasing;  the 
readings  on  the  upper  limb  will  be  made  first,  as  follows:  Set  the  index  on  an 
even  division  of  the  limb  at  a  reading  10'  or  15'  greater  than  the  double  altitude 
of  the  upper  limb.     When  the  two  images  are  then  brought  into  the  field  they 
will   appear   separated,  but  will   be   approaching   each   other.     The   observer 
watches  until  they  become  tangent,  when  the  time  is  carefully  noted  by  the 
chronometer.    The  index  is  then  moved  ahead  10',  15',  or  20',  and  the  same  pro- 
cess repeated.     A  little  practice  will  enable  the  observer  to  take  the  altitudes  in 
this  manner  at  intervals  of  10'  without  difficulty,  in  which  case  five  readings 
may  be  taken  which  will  correspond  to  an  increase  of  40'  in  the  double  altitude 
or  20'  in  the  actual  altitude.     As  the  sun's  diameter  is  about  32'  of  arc,  the  index 
may  now  be  moved  back  to  the  first  reading,  and  five  readings  on  the  lower 
limb  taken  at  the  same  altitudes  as  before.     In  this  case  the  images  will  overlap 
and  will  gradually  separate,  the  time  to  be  noted  being  that  when  the  two  disks 
are  tangent. 

If  the  sun  is  observed  west  of  the  meridian,  the  readings  on  the  lower  limb 
will  be  made  first.  The  altitudes  will  of  course  be  decreasing. 

113.  The  beginner  will  sometimes  find  difficulty  in  bringing  the  two  images 
into  the  field  together.     A  convenient  way  of  accomplishing  this  is  as  follows: 
Bring  the  index  near  the  zero-point  and  direct  the  telescope  to  the  sun,  when 
two  images  will  be  seen;  then  bring  the  instrument  down  towards  the  mercury 
horizon,  at  the  same  time  moving  the  arm  so  as  to  keep  the  reflected  image  in 
the  field  until  the  image  reflected  from  the  mercury  is  found,  when  both  will  be 
in  the  field  together.     A  little  practice  will  make  this  process  very  easy. 

In  observing  stars  care  must  be  taken  to  avoid  bringing  the  direct  image  of 
one  star  in  contact  with  the  reflected  image  of  another.  Sometimes  a  small 
level  is  attached  to  the  index-arm  to  facilitate  finding  the  reflected  image,  and 
at  the  same  time  for  preventing  mistakes  of  the  kind  just  mentioned.  It  may 
be  shown  geometrically  that  when  the  two  images  of  any  star  are  brought  in 
contact  in  the  manner  we  have  been  describing,  the  angle  formed  with  the 


194  PRACTICAL  ASTRONOMY.  §  I  14. 

horizon  by  the  index-glass  will  be  equal  to  that  formed  with  the  horizon-glass 
by  the  axis  of  the  telescope.  As  both  telescope  and  horizon-glass  are  fixed  to 
the  frame  of  the  instrument,  it  is  therefore  a  constant  angle.  If  then  the  level 
above  mentioned  is  adjusted  so  that  the  bubble  will  play  (the  plane  of  the  in- 
strument being  vertical)  when  the  index-glass  makes  this  constant  angle  with 
the  horizon,  it  may  be  used  for  the  purpose  mentioned.  The  method  of  finding 
the  reflected  image  will  then  be  as  follows:  Look  through  the  telescope  at  the 
image  reflected  from  the  mercury;  then,  holding  the  instrument  in  the  same 
position,  move  the  index-arm  until  the  bubble  plays.  If  the  reflected  image  is 
not  then  in  the  field  also,  the  reason  will  be  that  the  plane  of  the  instrument  is 
not  vertical.  It  will  be  brought  into  the  field  by  revolving  the  instrument  back 
and  forth  about  the  axis  of  the  telescope. 

To  adjust  this  level,  bring  the  two  images  of  the  sun  or  a  known  star  into  the 
centre  of  the  field  and  move  the  tube  until  the  bubble  plays. 


Errors  of  the  Sextant. 

114.  Among  the  various  theoretical  errors  to  which  sex- 
tant observations  are  liable  there  are  two  which  call  for  a 
detailed  investigation,  viz.,  index  error  and  eccentricity. 

To  Determine  the  Index  Error.  The  arc  is  graduated  a  short 
distance  backward  from  the  zero-point ;  when  the  reading 
falls  on  this  side  of  the  zero-point  the  reading  is  said  to  be 
off  arc  ;  a  direct  reading  being  on  arc. 

First  Method  of  Determining  the  Index  Error.  By  a  Star. 
Direct  the  telescope  to  a  star,  and  by  means  of  the  tangent- 
screw  bring  the  direct  and  reflected  images  into  exact  co- 
incidence. The  reading-  of  the  vernier  will  then  be  the  index 

o 

error,  and  it  must  be  applied  as  a  correction  to  all  angles 
measured  with  the  instrument. 

The  correction  will  be  -f-  when  the  reading  is  off  arc; 
The  correction  will  be  —  when  the  reading  is  on  arc. 

The  mean  of  several  readings  should  always  be  taken  so  as 
to  diminish  the  effect  of  errors  of  observation. 


§11$.  INDEX  ERROR.  1 95 

Example.  The  following  readings  were  made  with  a  Pistor 
&  Martins  sextant  for  determining  the  index  correction  : 

On  arc. 

45" 
60" 

70" 
70" 

75" 
60" 

30" 
75" 
7o" 
65" 

Mean  of  ten  readings,     i'  2//.o. 
The  index  correction  being  /,  we  have  therefore 

/==  --   l'2".Q. 

115.  Second  Method.  By  the  Sun.  Measure  the  apparent 
diameter  of  the  sun  by  bringing  the  direct  and  reflected 
images  tangent  to  each  other  and  read  the  vernier  ;  then 
bring  the  opposite  limbs  into  the  position  of  tangency  and 
again  read  the  vernier.  If  the  first  reading  is  on  arc,  the 
second  will  be  off  arc,  and  vice  versa. 

Let  r  =  the  reading  on  arc ; 

r'  =  the  reading  off  arc  ; 
/  ==  the  index  correction  ; 
5  =  the  true  diameter  of  the  sun. 

Then  5  =  r  +  /; 

S  =  r'  -  /• 
from  which  /=  4- ;>'  --  r) (209) 


196 


PRACTICAL  ASTRONOMY. 


§  1  1  6. 


When  observations  are  made  on  the  sun  for  any  purpose, 
the  gradual  heating  up  of  the  instrument  sometimes  changes 
the  value  of  the  index  correction.  For  this  reason  some  ob- 
servers determine  its  value  both  at  the  beginning  and  end  of 
such  a  series  of  observations.  The  following  example  taken 
from  the  Astronomische  Nachrichten,  Band  23,  No.  548,  will 
illustrate  this,  and  at  the  same  time  the  application  of  for- 
mula  (209)  : 


FIRST  DETERMINATION. 


On  arc. 

32'  20" 

20 

25 
20 


Off  arc. 

30'  60" 

60" 

50" 
50" 


r=32/2i".2     r'  =  30'  55 
/=*  -43".  i 


SECOND  DETERMINATION. 
On  arc.  Off  arc. 

31'  15" 

10" 


32'  5" 


O" 


20 
10 


=  32'  i".2     r'  —  31'  i3".S 
/  =  -23".; 


Eccentricity  of  the  Sextant. 

Il6.  As  the  arc  of  the  sextant  is  limited  and  is  read  by  a 
single  vernier,  the  effect  of  eccentricity  is  not  eliminated  ;  it 
should  therefore  be  investigated.  This  can  only  be  done  by 
comparing  the  values  of  angles  measured  by  it  with  their 
known  values  determined  in  some  other  way.  The  angles 
between  terrestrial  objects  may  be  measured  with  a  good 
theodolite,  and  the  same  angles  measured  with  the  sextant, 
or,  what  is  better,  stars  may  be  used. 

In  using  stars  for  the  purpose  \ve  may  proceed  in  either 
of  two  ways. 

First,  by  measuring  the  distances  between  known  stars.  The 
right  ascensions  and  declinations  of  the  stars  tvill  be  taken 
from  the  Nautical  Almanac  (it  will  be  best  to  use  none 
except  Nautical  Almanac  stars  for  the  purpose).  The  posi- 


§  Il6.  ECCENTRICITY  OF  SEXTANT.  197 

tions  of  the  stars  as  they  seem  to  us  will  differ  from  those 
given  in  the  Nautical  Almanac  by  the  amount  of  refraction 
in  a  and  8.  The  necessary  corrections  must  be  computed 
by  (194),  and  the  apparent  distances  of  the  stars  by  (IV)  or 
(IV),  Art.  67. 

Second,  by  measuring  the  altitudes  of  known  stars.  The  lati- 
tude of  the  place  of  observation  must  be  known  and  the  true 
time.  Then  from  (II),  Art.  65,  the  true  altitude  of  the 
star  may  be  computed,  or,  if  it  is  very  near,  the  meridian 
formula  (244)  may  be  used.  This  altitude  must  be  corrected 
for  refraction  to  make  it  comparable  with  that  measured  by 
the  sextant.  Whatever  plan  is  adopted,  the  angles  chosen 
should  be  such  that  the  measurements  will  be  distributed 
with  some  approach  to  uniformity  over  the  entire  arc  of  the 
sextant. 

Let    n'  =  the  value  of  the  angle  given  by  the  instrument ; 
n  =  the  true  value  of  the  same  angle ; 
z  —  the  correction  of  zero-point  for  eccentricity. 

Then  since  in  the  sextant  the  reading  of  the  arc  is  double 
the  actual  angle  passed  over  by  the  index-arm,  we  shall  have, 
from  formula  208, 

/  =  [»  -  (n'  —  *)]  =  2e"  sin  (fan  —  a)  ; 
and  for  the  zero-point,          z    =  —  2e"  sin  a. 

Subtracting,      n  —  n'  =  2/x[sin  (fan  —  at)  -f-  sin  ai]  ; 

irom  which        ;/  —  n'  =  4*"  sin  \n  cos  (J»  —  a).    .     .     (210) 

When  the  constants  e"  and  a  are  to  be  determined  from  ob- 
servation, equation  (210)  must   be    transformed  as  follows: 
Expanding  cos  (^n  —  a),  the  equation  becomes 

cos  at)  sin  \n  cos  \n  +  (\e"  sin  a)  sin2  \n  =  n  —  n'. 


198  PRACTICAL  ASTRONOMY.  §  1 1 6. 

Let    4e"  cos  a  =  x 

Ae"  sin  a  =  y 


z  =  the  sum  of  any  outstanding  constant  errors  ; 
sin  \n  cos  ±n  =  A,  a  known  coefficient  ; 
sin2  \n  =  B,  a  known  coefficient  ; 
n  —  ri  =  N,  the  quantity  given  by  observation. 

Then  each  measured  angle  gives  us  one  equation  for  deter- 
mining the  unknown  quantities  x,  y,  and  2,  viz., 


(212) 


If  everything  were  perfect,  three  such,  equations  would 
completely  solve  the  problem.  In  order  to  obtain  a  result  of 
practical  value,  however,  a  considerable  number  of  angles 
must  be  measured  and  the  resulting  equations  combined  by 
the  method  of  least  squares. 

Having  determined  x  and/,  we  have  e"  and  a  from  (211). 
With  these  values  a  table  of  corrections  is  then  to  be  com- 
puted by  (210). 

These  corrections  may  be  computed  for  intervals  of  10°, 
from  zero  up  to  the  largest  angles  ever  measured  with  the 
instrument.  The  correction  for  any  intermediate  point  may 
then  be  taken  out  by  interpolation. 

Example* 

We  give  as  an  example  the  investigation  of  the  eccentricity  of  sextant  "  Stack- 
pole  4152,"  made  by  Prof.  Boss  of  the  U.  S.  Northern  Boundary  Survey.  The 
observations  were  made  1873,  August  20,  at  the  U.  S.  Astronomical  Station 
No.  8. 

Latitude  =  <p  =  49°  i'  2''.  4  ;  determined  by  zenith  telescope.  f 
Longitude  =  L  —  ih  4im  18'  west  of  Washington. 

*  For  a  full  understanding  of  the  details  of  this  example  a  knowledge  is  re- 
quired of  some  principles  which  are  explained  later.  It  will  be  advisable  to 
read  Chapter  V.  before  attempting  it. 

f  See  Chapter  VIII. 


§  I  1 6.  ECCENTRICITY   OF  SEXTANT.  199 

Eleven  angles  were  carefully  measured,  each  measurement  consisting  of  ten 
readings.  All  except  two  were  measurements  of  double  altitudes  of  stars.  All 
were  north  stars  except  one,  viz.,  a  Aquila,  observed  on  the  meridian.  The 
north  stars  were  in  most  cases  observed  both  before  and  after  meridian  pas- 
sage ;  by  this  arrangement  any  small  undetermined  error  of  the  time  is  practi- 
cally eliminated. 

The  chronometer  correction  was  determined  by  measuring  the  altitudes  of 
a  Bootis  west  of  the  meridian  and  a  Andromeda  east,  both  being  observed  at 
exactly  the  same  altitude.* 

The  two  angles  which  form  the  exception  above  referred  to  were  measure- 
ments of  the  distances  between  a.  Andromeda  and  a  Pegasi,  and  a  Ursce  Minoris 
and  y  Cephei  respectively. 

The  index  correction,  determined  both  at  the  beginning  and  end  of  the  series, 
was  as  follows  : 

Beginning,     /  =  —  3'  43". 
End,  /  =  —  3'  42". 5. 

The  following  will  serve  as  a  specimen  of  the  form  of  record  and  method  of 
reduction.  The  series  of  ten  readings  is  divided  into  two  parts  so  that  one  may 
serve  as  a  check  on  the  other. 

Double  Altitude  of  a  Ursa  Majoris. 


Sextant. 

Chronometer. 

Sextan  f. 

Chronometer. 

I.  63°  25'  50" 

jgh   I2m  21* 

6.  62°  39'  45" 

I9h  I7m  l88 

2.            15    50 

13      23 

7.         29  50 

18      22 

3-  63     6  45 

14      23 

8.            21     10 

19     16 

4-  62   57    10 

,       15      21 

9-         13     5 

20      12 

5-         48  10 

16    20 

10.           3   55 

21        9 

Means                63°    6' 45"        igh  14™  2i8.6          62°  21' 33"  igh  19™  i58.4 

Chron.  correction                        A  T  —  22    50  .o  A  T  -T-  22    50.0 

True  time  =  9  =                     i8h  51™  3i8.6  i8h  56™  258.4 

From  ephemeris,  a.  =                     10   55    52  .o  10   55    52  .o 

Hour-angle  /  =                        7h  55m  398.6  8h    om  338.4 

t  =                    118°  54'   54"  120°    8'  21" 

The  true  altitude  of  the  star  at  the  instant  of  observation  is  then  computed  by 
formulae  (II),  Art.  65  : 

*  See  Articles  125,  126,  and  127. 


200 


PRACTICAL  ASTRONOMY. 


(p 

*  3 

t 

M 


h  = 
Refraction  r  = 


2ti 

Index  Cor.  / 
Computed  n 
Measured  n 


•-  49 

:  62° 
:Il8° 

-  75° 

;I5IC 
31° 

31° 
63° 


tan  d  =  0.282349 
cos  /  =  9.684407 


=  63 


Ol'  2 

26'  II 

54' 54 

50'    7". 4    tan  J/=o.597942n 

51'  g' 

38' 15 

29' 58 

i'30 
31' 28 

2' 57' 

3' 43 

6' 40' 

6' 45' 


tan  /  = 

cos  M  •=.  9.388649 
•  —  M}  =  0.085856 
tan  0  =  9. 7322 74n 

Proof  9.474505 

cos  d  —  9.665329 
cos  /  =  9 


n  —  n'  = 


9-349736n 
—  Ca.n((p—Af)=  0.157152,1 
.6      cos  a  —  9.944463,^  cos  a  =  9.944463^ 
'  tan  h  =  9.787311  cos  h  =  9.930768 


q>=    49°oi'o2". 

*8  =    62°  26'  1 1". 3      tan  6"  =  0.282349 

/  —  120°    8'  2i".o       cos  /  =  9.700792^ 
M—    75°i8'5o".i     tanAf  = 
<p—M=  124°  19'  52". 5 

a  —  152°    7'  51". 6 

h  —    31°    7'  i6".5 

r  =  i'  31". 7 

//'  =    31°    8' 48". 2 

Index  Cor.  I—  3' 43"  o 

Computed   «=    62°  21'  19". 4 
Measured   n  =    62°  21' 33". o 


tan  = 


n  —  n'  =     —  I3".6   cos  a  —  9.946462 
tan  h  —  9.780853 


Proof  9.474505 


tan  t  —  o.236i28n 
cos  M  =  9.404017 
>  —  M)  =  0.083130 

tan  a  =  g.'j232'jSn 

Proof     9.487147 

cos  d  =  9.665329 
cos  /  =  9.700792,1 


cos  a  = 

cos  h  =  9.932512 


Proof     9.487147 
Mean  =  N  =  —  g".i. 

The    computation    for    determining    the    true    angular    distance    between 


*  The  declination,  8,  is  taken  from  the  ephemeris. 


§  I  1 6.  ECCENTRICITY  OF  SEXTANT.  2OI 

a  Andromeda  and  a  Pegasi  is  also  given  in  full.     We  take  from  the  ephemeris 
for  1873,  August  20 — 

a  Andromeda  :  a  =    oh    im  51".  78      a  Pegasi  :  a  =  22h  s8m  28".  50 
8  =  28°    23'   30". 8  d  =  14°    31'   33".2 

The  observed  distance  was  20°    15'    20". 5 
Chronometer  time  2Oh  26m    3s.  6. 
Refraction  factor  B  X  t  X  T  —  .960.     [See  Eq.  (187).] 

We  first  determine  q  and  z  by  equations  (XII);  then  the  refraction  in  right 
ascension  and  declination  by  (194). 

a  ANDROMED^E. 
T  =  20h  26m    3s. 6 

A  7^  =  —    22    50  . 
0  =  20      3    13  .6 
a  =    o      i     51  .8 
t--    3"  58™  388.2 

/  =  —  59°  39'  33"        cos  /  =  9  70341        tan  t  =  o.23262n    cos  t  =  9.70341 
<p  =  49      i      2  .4    cot<£>  =  9.93890  cos  cp  =  9.81679 

N  =  23    41    39       tan  N  =  9.64231      sin  N  =  9.60407  9.52020 

d  =  28    23    31 
£-j-./V=52      5     10    sec  (d  -j-  N)  =    .21150        cot  =  9.89147 

q  =  —  48    10    21        «          tan  q  =    .O48ign  cosy  =  9.82405  cos  q  =  9.82405 
z  —  49    25    46  tan  z  =    .06742    sin  z  =  9.88059 

9. 70464 

9-81557  — Proof —  9.8155-6 

From  table,  mean  refraction  =  68".  i 

Factor  =  .960          Therefore  r  =  65". 4 

a  PEGASI. 

T  =      20h  26m    3s. 6 
AT  =         •     22    50  . 
0  =      20      3    13  .6 
a  =      22    58    28  .5 
*  =  -    2h  55m  i48-9 

'  —  —  43°  48'  44"        cos  /=  9.85830      tan    /=  g.gSiggn    cos  t=  9.85830 
q>  =      49      i       2  .4  cot  <p  =  9.93890  cos  q>  =  9.51679 

N  =      32      5      2      tan  A7"  =  9. 79720    sin  N  =  9.72523  9.67509 

d  =      14    31    33 

46    36    35sec(S+AO=    .16307     cot  =  9.97558 
—  36    34      5  tan  ^  =  9.87O29n  cos  =  9.90479  cos  q  —  9.90479 

49    39      o  tan  z  =    .07079   sin  z  =  9.882(,i 

9.78680 
9.88830  —Proof— 


202 


PR  A  CTICAL   A  S  TRONOM  Y. 


Mean  refraction  =  68". 6 
Factor  =.  .960 


Therefore  r  —  65". 9 


By  (194)— 


a.  ANDROMED/E. 
cos  q  =  9.82405 
log  r  =  1.81558 
sin  q  =  9.87225^ 
log,/£  —  i.63963n  tid  =  — 
cos  dda  =  1.68783 
15  cos  d  =  1.12043 
log  da  =    .56740    da  =  -\- 

C(Q  =        O 


a.  PEGASI. 


43"- 6 


cos  q  =  9.90479 
log  r  =  1.81889 
sin  q  =  9.77508,1 
log  dd  = 


i   dd  =  —       $2".g 

d0  =    28  23  30  .8     cos  dda  =  1.59397      £0  =  14  31  33  .2 
8=    282414.4     15  cos  S  =  1.16198       £—143226.1 
log  da  =    .43199    da  =  -f          2  .70 
«o  =  22  58  28  .50 


3  -69 
i  51  .78 
a  =     ohi'"488.09  a  =  22h58n'258.8o 

These  values  of  the  right  ascensions  and  declinations  of  the  stars  are  the  ones 
to  be  employed  in  computing  the  apparent  distance  between  the  two  stars  by 
equations  (IV)t. 


a'  = 

a  = 

24h 
22 

Im  48 
58  25 

'.09 
.80 

a'  -  a  = 

Ih 

3m  22s.  29 

a'  —  a  = 

15° 

50' 

34 

.35  cos  (a1  -  a)  = 

9.983181 

tan  (a1  —  a)  = 

9.452982 

d  = 

14 

32 

26 

.1       cot  d  — 

.586075 

sin  N  = 

9.984762 

N  = 

74 

54 

39 

.6      tanA^  = 

.569256 

sec  (A^+  d')  — 

.6376g8n 

d'  = 

28 

24 

14 

•4 

tan  B  = 

.075442n 

A/  1  /?' 

10*2 

18 

e  i 

o  cot  (JV  \  5'^  — 

-nf 

B  —  - 

1UJ 

49 

57 

04 

5 

.9      cos  £  = 

9.808504 

Proof 

.622460 

d  = 

20 

ii 

39 

.8       tan</  = 

9  565632 

1  = 

3 

43 

cos  (a'  —  a)  = 

9.983181 

n  = 

20 

15 

22 

.8 

cos  <5  = 

9.985862 

n  = 

20 

+'' 

15 

20 

2" 

•  5 

9.969043 

_ 

.3  —  <»« 

cos  B  — 

9.808504 

sin  d  = 

9.538079 

9.346583 

Proof 

.622460 

The  value  of  N  obtained  by  the  original  computation,  and  which  is  employed 
in  our  equations,  is  2". 2.  The  difference  is  of  no  importance  here. 

N  is  now  the  absolute  term  of  equation  (212).  For  the  coefficients  A  =  sin  %n, 
cos  ±n,  and  B  =  sin2  %n  we  must  employ  for  n  not  the  above  angles,  but  the  angle 
corresponding  to  the  point  on  the  limb  which  coincides  with  the  vernier-scale.  For 
example,  the  first  measured  angle  of  the  first  series  is  63°  25'  50".  The  limb 


=  69°  oo' 

=  69°  45' 

=  70°  oo' 

t 

=  70°  50' 

=  72°  15' 

—  72°  10' 

i«  =  17°  n'f       /sin 

=  9-47075 

=  63°  30' 

/cos 

=  9.98014 

=  65°  15' 

A  =  0.2824        log  A 

=  9-45089 

§  1 1 6.  ECCENTRICITY  OF  SEXTANT.  203 

was  graduated  directly  to  10';  these  intervals  were  subdivided  by  the  vernier  to 
10".  The  zero-point  of  the  vernier  falls  between  63°  20'  and  63°  30';  then  read- 
ing along  the  vernier  to  the  point  where  coincidence  takes  place,  we  find  this  to 
be  at  the  reading  69°  10'  of  the  limb.  It  is  therefore  the  eccentricity  of  this 
point  by  which  our  angle  is  affected,  and  not  that  of  the  point  63°  25'  4~. 

In  this  way  we  find  the  point  of  contact  for  each  reading  of  our  series  as 
follows  : 

63°' 25'  50"  Point  of  contact  =  69°  10' 

6'  45" 

62°  57'  10" 

48'  10" 

39'  45" 

29'  5o" 

2l'   10" 

13'    5" 
62°    3' 55"  =  65°  55'        Z?  =  0.0874        log  B  =  8.94150 

Mean  =  n  =  68°  47' 

Therefore  from  this  series  we  derive  the  equation 

0.2824*  4~  0.08747  4~ z  —  ~~  9".!. 

By  proceeding  in  a  similar  manner  with  each  of  the  eleven  angles  measured, 
the  following  equations  of  condition  are  obtained  : 

.0703*  4-  .00507  -f  z  =-  5.5; 

.1104*  4-  .012374-  z  —  -f-  2.2; 
.2019*  4-  .04257  4-  z  —  —  7.3; 
.2341*  4-  .05827  4-  z  =  —  17.5; 
.2824*  4-  -087474- z  —  —  9.1; 

.3295*  4-  .1239^  -+-  z  =  —  18.5; 
.3586*4-  .151574-  z  =  —  10.5; 
•3933*  +  -I9I3/  4-  «  =  —  14-0; 
.3997*  -j-  .19967  4-  z  =  —  24.0; 

.4244*  4-  .23577  +  z  =  -  46.2; 

.4423*  4-  .26687  +  *  =  —  28-°« 

It  will  be  seen  that  the  coefficients  of  *  and  7  are  much  smaller  throughout 
than  those  of  z,  while  the  absolute  terms  are  relatively  large.  It  would  there- 
fore be  a  little  more  systematic  to  render  the  equations  homogeneous,  as  ex- 


204 


PRACTICAL  ASTRONOMY. 


§116. 


plained  in  Art.  24,  before  forming  the  normal  equations.     This  has  not  been 
done,  however. 

The  details  of  the  formation  of  the  normal  equations  (Articles  21  arfd  25)  are 
as  follows  :    As  the   number  of  unknown  quantities  is  three,  we  rule  our  sheet 

into  — ~ —  i  =  14  vertical   columns  (Art.  25),  to  which  we  have 

added  two  columns  for  the  residuals  (v)  and  their  squares  (vz>).     These  will  be 
filled  in  after  the  unknown  quantities  have  been  determined. 


No. 

ac 

be 

cc 

en 

cs 

tut 

ab 

an 

i 

.0703 

.0050 

-   5-5 

6-5753 

.00494 

.00035 

-   -3867 

2 

.1104 

.0123 

+    2.2 

—  1-0773 

.01218 

.00136 

+   -2429 

3 

.2019 

.0425 

—   7-3 

8.5444 

.04074 

.00859 

—  !-4739 

4 

•234l 

.0582 

—  17  5 

18.7923 

.05480 

.01362 

—  40967 

5 

.2824 

.0874 

-   9-i 

10.4698 

.07976 

.02468 

-  2.5698 

6 

•3295 

•  239 

-  18.5 

J9  9534 

.10859 

.04084 

—  6.0957 

7 

.3386 

•  515 

-  105 

12.0101 

.12854 

.05432 

—  3  7653 

8 

•3Q33 

•  913 

•-  14.0 

T5-5846 

•15467 

.07522 

-  5-5062 

9 

•3997 

•  996 

—  24.0 

25-5993 

•15974 

.07976 

—  9.5928 

JO 

•4^44 

-  46.2 

47-8601 

.18014 

.  1OOU2 

—  19  6073 

ii 

.4423 

.  668 

-  28.6 

30.3091 

.19561 

.IlSoi 

-  12  6498 

32469 
[ac] 

1-3742 
\bc\ 

11.  0 

[cc] 

-  179.0 
[en] 

I94.62II 

[cs] 

I  11971 

[aa] 

•5l677 

M 

-  65-5013 

[an] 

No. 

as 

bb 

bn 

bs 

nn 

ns 

V 

vv 

i 

-f   .4622 

.00002 

-   -0275 

+  .0329 

3025 

-   36-16 

+  i.7 

2.89 

2 

—   .1189 

.00015 

T   -027! 

—  .0133 

484 

—    2.37 

-  6.1 

37-2i 

3 

+  *  7251 

.00181 

—   3T°3 

+  -3631 

53-29 

—   62.37 

+  10 

.  1  OO 

4 

4-3993 

.00339 

—  1.0185 

L0937 

306.25 

-  32887 

+  9-7 

94.09 

5 

2.9567 

.00764 

-   -7953 

•9I5I 

82.81 

-   95-28 

—  1.9 

3.61 

6 

65746 

.01536 

—  2.2922 

2.4722 

342-25 

-  369'4 

+  3-2 

10.24 

7 

4.3068 

.02296 

—  I-5907 

i  8195 

110.25 

—  126  ii 

-   8.2 

67.24 

8  • 

6.1294 

•03657 

—  2.6782 

2.9813 

196.00 

—  218.18 

-  9.8 

96.04 

9 

10.2319 

.03982 

—  4.7904 

5-1096 

576.00 

-  6,4.38 

-  0.9 

0.81 

10 

20.3118 

•05554 

—  10.8893 

11.2806 

2134.44 

—  2211.14 

-f  16.6 

275  56 

ii 

13  4057 

.07118 

-  7-6305 

8.0865 

817.96 

-  86684 

-  5.2 

27.04 

70.3846 
[as] 

w 

-r*r 

341412 

[6s] 

4654-34 
[nn] 

—  4930-84 
[ns] 

61573 

[vv] 

The  correctness  of  the  work  up  to  this  point  is  now  verified  by  substitution 
in  proof-formula  (44). 

Therefore  the  normal  equations  are  as  follows  : 

1.1197*+    .5168}'+    3.24692=—    65.5013; 

.5i68*-|-    .25447 -f     1.37422  =  -    3L9958; 

3.2469*  -j-  1.37427  -}-  n.ooooz  =  —  179.0000. 


§  1 1 6.  ECCENTRICITY  OF  SEXTANT.  2O$ 

For  the  solution  of  these  equations  we  make  use  of  the  form  given  in  Art.  32. 


[aa]        1.1197 
/  =  0.049102 

[«£]         .5168 
'  =  9-7I3323 

[a.:]  =     3.2469 
/=    05x1469 

[an]  —65.5013 
/  =1.8162500 

[as]     70.3846 

/«:    1.847478 

E 

I' 
E 

II 
III' 

'[^—  ' 

[AA]        .2544 
•2385 

[be]        1.3742 
1.4986 

[An]  -31.9958 
-30.2323 

[6s]      34.1412 

32.4862 

,M=  0.46,367 

[AA  i]        .0159 
/  =  8.20140 

[be  ij     —  -!244 
/  =    9  094820 

[Am]—  1.7635 
/  =  0.24638,! 

[6si]        1.6550 
/  =  o  21880 

[cr]  =  11.0000 
9-4I53 

[en]  —179.0000 
—  189.9403 

[fj]        194.6211 
204.1009 

'M-°*»* 

[<T6-i]=    1.5847 
9733 

[cm]  =10.9403 

[csi]  =  —  9.4798 

—  12.9485 

[f<72]  =      .6114 
/=    9.78633 

[en  2]—  2.8571 
/  =  0.45592,, 

[CS2]         +    3.4687 

lz=    o.66959n 

z  =  —  4".  673 

/gj  =  ,,67I48 

[nn]  =  4654-34 
383^-76 

[«j]=  -4930.84 
~4II7-43 

Proof-Form  ulce. 
V  .  [Asj.]=  \bb-i\-\-\bcT\—  \bni\i 

[An,] 

[nni]=    822.58 
195-59 

[ns  i]  -  813.41 
-   183-56 

wt         "•   t« 
VI1      IIP.   [cs 

VII.     «* 
VIII.     w 
VIII     IX.     «f 

The  wo 
rious  stag 
IX     or  all  of  tl 

2]=  [eC2]  —  [en  2] 

-[c«i]; 

![AAi]    -3-°449"n 

[«I2] 

\nnz\=    626.99 
13-35 

[ns  2]  —  629.85 
—     16.21 

3]=  —  [»»s]- 

'k  is  checked  at  the  va- 
;s  by  substitution  in  any 
ic  above  proof-formulae. 

'&*«]     "°'66959n 

>«3]=    613.64 

[ns  3]  -  613.64 

The  elimination  equations  (56)  are  here  rewritten  for  convenience: 

\aa~\x  -f-  [aH]  y     -f"  [ar]0      =  [«;/]  ; 
[^  i]j+  [be  i]ar  =  [*»  i]. 

By  substituting  in  these    the  coefficients,  the  logarithms  of  which  are  in  the 
horizontal  lines  marked  E  in  the  foregoing  scheme,  we  find 


y  =  -  I47"-47I 


=  -f  23".I2. 


These  values  substituted  in  the  equations  of  condition  give  the  residuals  v. 
For  the  final  proof  of  the  accuracy  of  the  entire  computation  we  have,  Eq.  (62), 

[nn  3]  =  [vv\. 

The  agreement,  though  not  exact,  is  sufficiently  close  for  our  purpose,  and  as 
close  as  could  be  expected  when  the  magnitude  of  some  of  the  numerical  quan- 
tities involved  in  the  equations  is  considered. 


2O6 


PRACTICAL  ASTRONOMY. 


§116. 


For  determining  the  weights  of  x,y,  and  zwe  employ  equations  (76),  by  means 
of  which  we  find 

pf    =         .6114;  py    =         .006135;  px    =         .01196. 

The  mean  error  of  an  observation  we  obtain  by  formula  (88),  viz., 


e  —  ± 


l-  =  8".7725. 


m  -  3 
The  mean  errors  of  x,  y,  and  z  are  then  given  by  equations  (89): 


=  —  =.  =  80  .21; 


e  e 

—  —  —  =  H2  .00;         ez  =  —  —  =  Ii".22. 


These  quantities  multiplied  by  .6745  give  the  probable  errors. 

Collecting  our  results,  we  have  the  following  values  of  x,  yt   z,   with  their 
probable  errors: 

x  -  +  23".!  ±  52".9; 
y  =  —  i47"-5  ±  75"-5; 
z  —  —  4"-7  ±  7"-6. 

We  next  compute  a  table  of  corrections,  to  be  employed  with  this  instrument, 
by  formulae  (211)  and  (210),  viz.  : 

4e"  cos  a  =  x; 
4/'  sin  a  =  y\ 
n  —  n'  =  4f"  sin  ±n  cos  ($n  —  or). 


We  find 


4<r"  =  149".  3;         a  =  —  81°  6'. 


Substituting  for  n  successively  10°,  20°,  etc.,  we  have  the  following  table  of 
corrections : 


Angle. 

Correction. 

Angle. 

Correction. 

0° 

o".o 

80° 

^"V'T 

10° 

+  o".7 

90° 

-  i3"-4 

20° 

+  o"-9 

100 

-  17'  -5 

30° 

+  o".5 

110° 

—  22'  .0 

40° 

-  of'.s 

120° 

-  26'  .9 

50° 

-  2".0 

I300 

—  32'  .1 

60° 

-  4".  I 

140 

-  37'  -7 

70° 

-  6".7 

§  I  I /.  OTHER  ERRORS   OF  SEXTANT.  2O/ 

Other  Theoretical  Errors. 

117.  In  a  complete  theoretical  discussion  of  the  sextant  there  are  several 
other  sources  of  error  which  require  consideration.  The  more  important  of 
these  are  the  following:  prismatic  form  of  the  index-glass,  of  the  colored  glass 
shades \  and  of  the  horizon-roof;  want  of  perpendicularity  of  the  planes  of  the  in- 
dex and  horizon  glass  to  the  plane  of  the  instrument;  inclination  of  line  of  collima- 
tion  of  telescope  to  plane  of  instrument;  errors  of  graduation  of  the  limb. 

With  a  good  instrument  well  adjusted  the  effect  of  any  one  of  these  will  be 
small,  although  they  may  combine  together  in  such  a  way  as  to  produce  a  very 
appreciable  effect  on  the  value  of  a  measured  angle.  Not  much  can  be  gained, 
however,  practically  by  investigating  in  detail  the  forms  of  the  corrections  re- 
quired. The  experienced  observer  will  avoid  these  errors  as  far  as  can  be 
done  by  careful  adjustment,  and  then  will  arrange  his  observations  with  a  view 
to  eliminating  from  the  results  such  of  them  as  remain  undetermined.  See 
Art.  127. 


The  Chronometer. 

118.  The  chronometer  is  simply  a  watch  made  with  special 
care,  and  in  which  the  balance-wheel  is  so  constructed  that 
changes  of  temperature  will  produce  the  least  possible  effect 
on  its  time  of  oscillation.  The  test  of  a  good  chronometer 
is  the  uniformity  of  its  rate  from  day  to  day.  It  is  impossi- 
ble to  make  an  instrument  so  perfect  that  24h  as  shown  by  it 
shall  exactly  correspond  to  one  day,  but  its  excellence  is  in- 
dicated by  the  uniformity  with  which  it  gains  or  loses. 

The  daily  rate  of  a  chronometer  is  the  amount  which  it 
gains  or  loses  in  24  hours. 

The  error  of  the  chronometer  \§  the  difference  between  the 
time  as  shown  by  the  face  of  the  instrument  and  the  true 
time.  ^ 

The  chronometer  correction  is  the  amount  which  must  be 
added  to  the  reading  of  the  chronometer-face  at  any  instant 
to  give  the  true  time;  it  is  equal  to  the  error  with  its  sign 
changed. 

It  is  a  convenience  to  have  the  error  and  rate  small,  but  it 


208  PRAC7^ICAL   ASTRONOMY.  §  I  1 8. 

is  not  essential.  Chronometers  are  made  in  two  different 
forms,  viz.,  box-chronometers  and  pocket-chronometers. 
The  first  form  of  instrument  is  generally  suspended  by 
means  of  gimbals  in  a  wooden  box,  in  such  a  manner  that, 
whatever  the  position  of  the  box,  the  face  of  the  instrument 
will  maintain  a  horizontal  position.  This  arrangement  is 
useful  at  sea,  but  for  transportation  on  land  the  instrument 
must  be  securely  fastened,  as  otherwise  the  violent  agitation 
produced  by  sudden  shocks  would  be  injurious.  The  bal- 
ance-wheel of  this  form  of  instrument  oscillates  at  half-second 
intervals. 

The  pocket-chronometer  is  generally  somewhat  largei 
than  an  ordinary  watch.  The  oscillation  or  beat  is  a  little 
more  rapid  than  with  the  box-chronometer;  thus  the  pocket- 
instruments  of  T.  S.  and  J.  D.  Negus  beat  live  times  in  two 
seconds. 

A* chronometer  regulated  to  sidereal  time  is  more  conven- 
ient for  observation  on  stars.  With  the  sun  a  mean  time 
chronometer  is  preferable. 

The  error  and  rate  will  be  considered  more  fully  in  con- 
nection with  the  subject  of  determining  time.  Most  chro- 
nometers require  winding  every  24  hours.  This  should  be 
done  at  about  the  same  time  each  day,  as  if  they  are  al- 
lowed to  run  much  longer  than  the  usual  time  a  different 
part  of  the  spring  comes  into  action,  which  may  affect  the 
rate.  Such  instruments  will  run  for  48h  or  more  before 
stopping,  so  that  in  case  the  winding  should  be  neglected 
for  one  day  they  will  be  found  running  the  next;  but  for 
the  reason  just  stated  this  should  not  occur. 

Comparison  of  Chronometers. 

119.  When  the  errors  of  several  chronometers  are  to  be 
determined  at  the  same  time,  the  error  of  one  of  them  is  ob- 


§119-  THE    CHRONOMETER.  2OQ 

tained  by  observation,  and  of  the  others  by  comparison  with 
this.  When  two  sidereal  or  two  mean  solar  chronometers 
are  compared  together  the  beats  will  be  sensibly  of  the  same 
length,  but  generally  the  two  will  not  beat  exactly  together; 
the  fraction  of  a  second  by  which  the  beat  of  one  falls  be- 
hind that  of  the  other  must  therefore  be  estimated.  With 
some  practice  this  can  be  done  so  that  the  error  in  the  esti- 
mation will  not  much  exceed  os.i. 

When  a  sidereal  is  to  be  compared  with  a  mean  time  chro- 
nometer the  error  of  comparison  will  be  much  smaller. 
Since  Is  of  sidereal  time  is  equal  to  o".9972/  mean  solar  time, 
it  follows  that  the  sidereal  gains  os.oo273  on  the  mean  time 
chronometer  in  one  second;  this  gain  will  amount  to  one 
entire  beat,  or  os.5,  in  183*,  or  approximately  3™.  Therefore 
practically  once  every  three  minutes  the  beat  of  the  two  will 
coincide.  It  is  found  that  with  a  little  practice  the  ear  can 
detect  a  discordance  in  the  beats  as  long  as  they  differ  by 
os.O2  or  os.O3,  and  therefore  the  comparison  can  be  made 
within  this  limit  of  error. 

When  a  number  of  chronometers  are  to  be  compared  with 
a  standard  clock,  it  may  be  done  very  conveniently  by  means 
of  the  chronograph.*  The  clock  being  connected  with  the 
chronograph,  the  observer  taps  the  signal-key  in  coincidence 
with  one  or  more  even  beats  of  the  chronometer,  and  thus 
the  time  by  both  clock  and  chronometer  are  recorded  on 

the  same  sheet. 

\ 

i 

The  Astronomical  Clock.  , 

120.  In  a  fixed  observatory  the  clock  is  an  instrument  of 
great  importance.  It  is  generally  regulated  for  sidereal 
time.  The  only  part  of  the  mechanism  which  requires  notice 

*  See  Art.  121. 


2IO  PRACTICAL  ASTRONOMY.  §  I2O. 

here  is  the  pendulum,  which  is  made  of  the  necessary  length 
to  beat  seconds. 

The  rate  of  the  clock  depends  upon  the  length  of  the 
pendulum;  and  since  a  rod  of  metal  changes  its  length  with 
every  change  of  temperature,  some  method  of  compensation 
is  necessary  in  order  to  keep  the  centre  of  oscillation  at  a 
constant  distance  from  the  point  of  suspension.  For  accom- 
plishing this  two  different  forms  are  used,  viz.,  the  gridiron 
and  the  mercurial  pendulum. 

In  the  gridiron  pendulum  the  rod  is  composed  of  a  num- 
ber of  parallel  bars,  alternately  of  brass  and  steel.  These  are 
so  arranged  that  the  expansion  of  the  steel  bars  tends  to  in- 
crease the  length,  while  that  of  the  brass  bars  tends  to  dimin- 
ish it.  As  these  metals  expand  and  contract  by  different 
amounts  when  subjected  to  changes  of  temperature,  the 
relative  lengths  of  the  two  may  be  so  adjusted  as  to  maintain 
a  constant  length  for  the  system. 

With  the  mercurial  pendulum  the  rod  consists  of  a  single 
bar  of  steel.  The  "  bob"  is  a  cylindrical  vessel  of  glass  or 
metal  filled  with  mercury.  The  expansion  of  the  rod  de- 
presses the  centre  of  oscillation,  while  that  of  the  mercury 
raises  it.  Thus  by  making  the  cylinder  of  proper  propor- 
tions, as  compared  with  the  rod,  the  necessary  compensation 
is  effected. 

With  a  clock  which  is  exposed  to  sudden  changes  of  tem- 
perature the  gridiron  pendulum  will  give  a  more  uniform 
rate  than  the  mercurial,  as  the  comparatively  thin  bars  of 
metal  will  accommodate  themselves  to  the  temperature  of 
the  air  much  sooner  than  the  comparatively  large  mass  of 
mercury. 

The  density  of  the  air  as  indicated  /by  the  barometer  also 
affects  the  rate  of  the  clock  by  its  variable  resistance  to  the 
motion  of  the  pendulum.  Struve  found  for  the  standard 
clock  of  the  Poulkova  observatory  a  change  of  os-32  in  rate 


§121.  THE    CHRONOGRAPH.  211 

for  a  variation  of  one  inch  in  the  barometer.  It  is  therefore 
very  important  to  protect  the  standard  clock  from  sudden 
and  extreme  atmospheric  changes.  In  some  observatories 
this  is  done  by  placing  it  in  an  air-tight  compartment  below 
the  surface  of  the  ground. 


The  Chronograph. 

121.  The  chronograph  is  used  in  connection  with  the  clock 
for  registering  graphically  on  a  strip  or  sheet  of  paper  the 
beats  of  the  latter.  Fig.  22  shows  a  common  form  of  this 
instrument.  The  sheet  of  paper  on  which  the  record  is  to 
be  made  is  wrapped  around  the  cylinder,  which  in  this  in- 
strument is  14  inches  long  and  6  or  7  inches  in  diameter. 
The  cylinder  is  given  one  revolution  per  minute  by  means 
of  the  clockwork.  The  pen  which  is  shown  above  the  cyl- 
inder is  supplied  with  aniline  ink,  and  being  moved  slowly 
along  in  the  direction  of  the  axis  of  the  cylinder  it  traces  a 
continuous  spiral  on  the  surface. 

The  apparatus  is  placed  in  an  electric  circuit  passing 
through  the  clock,  and  so  arranged  that  the  pendulum  breaks 
the  circuit  for  an  instant  at  the  beginning  of  each  second.* 
By  means  of  a  spring  which  acts  in  the  direction  contrary 
to  that  of  the  electro-magnet  shown  in  the  figure,  the  pen  is 
thus  given  a  slight  lateral  motion  at  each  beat  of  the  clock, 
producing  instead  of  a  continuous  line  a  line  graduated  as 
shown  in  the  folding  plate,  Fig.  220. 


*  The  arrangement  may  be  such  that  the  circuit  will  be  closed  for  an  instant 
at  the  beginning  of  each  second,  remaining  open  during  the  remainder.  The 
break-circuit  plan  is  the  one  more  commonly  employed.  Various  mechanical 
devices  are  employed  by  different  makers  for  causing  the  clock  to  open  or  close 
the  circuit. 


212 


PRACTICAL   ASTRONOMY. 


Fi 


§121.  THE   CHRONOGRAPH.  21$ 

Each  of  these  spaces  is  the  graphic  record  of  one  second  of 
time  as  shown  by  the  clock.  The  beginning  of  the  minute  is 
marked  by  the  omission  of  one  of  the  points.  The  instru- 
ment here  shown  will  run  2-J-  hours.  When  the  paper  is 
removed  from  the  cylinder  and  spread  out  it  is  marked  with 
parallel  lines,  each  line  being  the  record  of  one  minute  of 
clock  time. 

In  order  to  make  use  of  this  apparatus  for  recording  the 
time  of  the  occurrence  of  any  phenomenon,  the  wire  which 
forms  the  circuit,  passing  from  the  battery  through  the 
clock  and  chronograph, is  made  to  pass  through  a  signal-key 
held  in  the  hand  of  the  observer,  and  by  means  of  which  the 
circuit  can  be  instantly  broken. 

In  Fig.  23,  aa'  is  the  wire  through  which  the  circuit  passes. 
When  the  point  b  touches  the  metallic  plate  c  r> 

the  circuit  is  closed.  A  key  is  so  arranged 
that  by  tapping  it  with  the  finger  this  point 
is  raised  and  the  circuit  broken;  this  pro-  *  FIG. 93. 
duces  a  mark  on  the  chronograph-sheet  similar  to  that  made 
by  the  clock,  and  the  position  of  which  is  the  record  of  the 
instant  when  the  key  was  pressed. 

Fig.  22a  is  a  reduced  copy  of  the  chronograph  record  of 
transits  of  the  stars  6  Aquarii ,  y  Aquarii)  7t  Aquarii,  6  Aquarii, 
a  Lacerta,  and  ri  Aquarii  observed  with  the  transit-circle  of 
the  Washington  observatory,  1884,  December  7. 

Each  star  is  observed  over  eleven  threads.*  The  record 
begins  by  striking  the  signal-key  several  times  in  quick  suc- 
cession before  the  star  reaches  the  first  thread,  in  order  to 
mark  the  beginning  of  the  series;  then  it  is  tapped  in  exact 
coincidence  with  the  star's  passage  over  each  thread  in  suc- 
cession. 


*  See  Art.  170. 


a 


214  PRACTICAL  ASTRONOMY.  §  121. 

Taking  the  first  of  the  above  stars,  6  Aquarii,  our  chrono- 
graph-sheet gives  the  following  record  : 

22h  iom  33s-4  22h  iom  47s-9 

36s.o  5o8.o 

37S.6  54s. i  , 

4is.7  558-7 

43S.8  22h  iom  588.3 

22h  iora  45S.8 

For  reading  the  record  a  scale  long  enough  to  reach  the 
entire  length  of  the  sheet  is  used,  the  spaces  of  which  are 
the  same  as  those  of  the  sheet.  These  spaces  are  numbered 
continuously  from  o  up  to  60;  each  space  being  divided  to 
tenths,  the  fractional  parts  of  these  subdivisions  may  be  esti- 
mated. 

While  the  paper  is  on  the  cylinder  it  is  necessary  to  mark 
somewhere  on  the  sheet  the  hour  and  minute  shown  by  the 
clock  ;  this  serves  as  a  starting-point  for  reading  the  record. 

For  the  purpose  of  determining  longitude,  chronometers 
are  sometimes  provided  with  a  break-circuit  attachment, 
when  they  can  be  used  with  a  chronograph  in  the  same  man- 
ner as  a  clock. 

The  main  advantages  which  the  chronograph  possesses 
over  the  methods  employed  before  its  introduction  are, 
first,  a  comparatively  inexperienced  observer  can  record 
astronomical  phenomena  by  its  use  with  a  degree  of  accuracy 
which  it  would  take  months  or  perhaps  years  of  practice  to 
acquire  without  it ;  and  second,  the  record  is  made  by  simply 
pressing  a  key  with  the  finger :  thus  many  more  observations 
can  be  made  in  a  given  time  than  is  possible  when  everything 
must  be  written  down  with  a  pencil 


CHAPTER  V. 

DETERMINATION   OF  TIME  AND   LATITUDE.— METHODS 
ADAPTED   TO   THE    USE   OF   THE   SEXTANT.* 

122.  In  a  spherical  triangle,  when  three  parts  are  known  any 
other  part  may  be  determined.     Let  us  consider  the  triangle 
PZS,  where  P  is  the  pole  of  the  heavens,  Z  p 

the  observer's  zenith,  and  5  a  known  star 
(the  word  star  here  including  the  sun,  moon, 
or  a  planet). 

If  we  measure  the  altitude  of  5,  the  side 
SZ  of  our  triangle  is  known.  The  declination 
d  is  taken  from  the  Nautical  Almanac.  If 
then  we  know  the  hour-angle  /,  we  have  the 
data  for  determining  the  latitude  (p.  If  (p  is 
known,  we  have  the  hour-angle  t  by  compu- 
tation, and  therefore  the  true  local  time,  from 

(197). 

We  have  then  simply  to  give  the  solutions  of  this  triangle 
best  adapted  to  the  different  cases  which  will  be  considered, 
and  to  determine  what  conditions  will  be  most  favorable  to 
accuracy. 

Determination  of  Time. 

123.  By  a  single  altitude  of  the  sun. 

Let  h'  —  the  observed  altitude  of  the  sun's  limb,  corrected 
for  index  error ; 


*  The  methods  of  this  chapter  are  of  course  equally  adapted  to  the  use  of  any 
instrument  for  measuring  altitudes. 


PRACTICAL   ASTRONOMY.  §  124 

h  —  the  true  altitude  of  the  sun's  centre  ; 
z  —  the  true  zenith  distance  of  the  sun's  centre  =  90°—  h\ 
r  =  the  correction  for  refraction  ; 
p  =  the  correction  for  parallax  ; 
s  =  the  correction  for  semidiameter. 

Then  k  =  h'  --  r  +  p  ±  s  ......     (213) 

s  is  ±  when  the  j  ^JI^.  [   limb  is  observed. 


The  required  solution  of  the  triangle  may  now  be  deduced 
from  the  last  of  equations  (121),  viz., 

cos  2  =  sin  cp  sin  d  -f-  cos  (p  cos  d  cos  t  ; 
from  which 

cos  z  —  sin  (p  sin  d 
cos  /  =  -  —  «  --  .     .     .     .     .     .     .     (214) 

cos  cp  cos  6 

In  some  cases  this  equation  may  be  conveniently  employed 
for  computing  •*,  as  when  the  same  star  is  observed  on  several 
successive  days  at  the  same  place,  sin  cp  sin  d  and  cos  (p  cos  d 
may  then  be  considered  constant  for  a  week  or  more  in 
ordinary  sextant  work.  The  numerator  will  be  computed 
with  addition  and  subtraction  logarithms. 

As  t  is  given  in  terms  of  the  cosine,  this  equation  should 
not  be  used  when  the  angle  is  less  than  45°. 

124.  To  place  (214)  in  a  form  more  generally  applicable, 
first  subtract  both  members  from  unity,  then  add  both  mem- 
bers to  unity,  viz.: 

cos  cp  cos  d  4-  sin  cp  sin  d  —  cos  z 
I  —  cos  t  —  - 


I  -\-  COS  /  = 


cos  cp  cos 

cos  cp  cos  #  --  sin  cp  sin  S  -f-  cos  z 
cos  cp  cos  d 


§124.  DETERMINATION   OF    TIME.  21 J 

from  which  we  easily  obtain 


-V 


sin  j[>  +  (y  —  <?)]  sin  £|>  —  (<p  -    6)\ 

cos  <z>  cos  6  *     ^2I5> 


cos      g  cos         - 


-_         sn  y  -  c)    sn      ^  -    y>  - 

:   V   COS  i|>+  (^  +  0)]  COS  fa  -(9+  *)]' 

For  most  purposes  equation  (215)  will  give  the  necessary 
degree  of  precision. 

When  the  extremest  accuracy  is  required  (217)  should  be 
used. 

These  equations  give  t  in  degrees,  minutes,  and  seconds  of 
arc.  For  our  purposes  it  must  be  reduced  to  time  by  divid- 
ing by  15. 

Then  let  T0  —  the  chronometer  time  of  observation; 
AT  =  the  chronometer -correction  ; 
E  =  the  equation  of  time. 

Then  the  apparent  time  of  observation  is  t  (Art.  90). 

Mean  time  of  observation      =  t  -\-  E  =  T0  +  AT\  \       ,    ^ 
from  which  AT  =  t  +  £  —  T0.  } 

AT  is  the  quantity  required. 

In  the  above,  where  the  object  observed  was  the  sun,  we 
have  supposed  the  chronometer  to  be  regulated  to  mean  time. 
If  a  sidereal  chronometer  has  been  used,  the  mean  time 
(/  -|-  E}  must  be  converted  into  sidereal  time  by  (200)  or  (201) 
and  the  resulting  value  compared  with  the  chronometer  time. 


218  PRACTICAL   ASTRONOMY.  §  124. 

Example  I.  West  Las  Animas. 
Observation  of  sun  for  time. 

Sextant.  Chronometer. 

0  88°  50'  00"  3h  35'"  12s. 

89   °°     o  35     39  .5 

10     o  36       3  .5 

20     o  36     30.5 

89    30     o  36     56  .5 

088°  50'    o"  3h37m55'.5 

89       O       O  38      22  . 

10       O  38      48  . 

20     o  39     14  .5 

89    30     o  39     41  .o 


Means       89°  10'    o"  3h  37™  26". 3 

/  —  ii 

Eccentricity  —  45 


zA  =  89°    9'    4' 

A  =  44    34   32 

Refraction  r  =  —  49 

Parallax/  =  6 


h  =  44°  33'  49" 

z  =  45    26    ii     =  zenith  distance  of  sun's  centre. 

We  have  now  the  data  for  applying  formulae  (215)  and  (218). 

.  A* 

<p  =  38°    4'    o"  sec  =  0.10386  9.9 

d  =  18    42    17  sec  =    .02357  4-3 

cp  —  d  =  19    21    43 
z  =  45    26    ii 
z  -{-  ((p  —  d)  —  64    47    54 
s  —  (q>  —  d)  =  26      4    28 

i[z  +  (<p  -  <5)]  =  32    23    57     =  S  .  sin  =  9.72901  19.9 

|[*  —  (<p  —  S)\  =  13      2    14     =  D  .  sin  ='  9.35331  54.6 

sin2  it  =  9.20975 

it  =  23°  44'  28"  sin  it  =  9.60487  28.7 

/  =  47    28    56 
/=  -    3h    9m  55"- 7 

/  =          20     50        4.3 

E  =  -f  6  13  .o 

/-}-£"=        20  56  17  .3  =  mean  solar  time. 

T  =         3  37  26  .3  =  observed  time. 

4T=—    6  41       9  .o  =  chron.  correction  [Eq.  (218)]. 

This  value  differs  but  little  from  the  value  assumed  above.  If  the  difference 
had  been  large  it  would  have  been  necessary  to  take  from  the  ephemeris  the 
value  of  d  for  this  more  correct  time,  and  to  repeat  the  computation  for  a  more 
correct  value  of  A  T.  Or,  if  the  difference  were  not  too  great,  the  necessary 
correction  could  be  determined  by  a  differential  formula. 

*  These  values  are  written  down  for  the  purpose  of  computing  the  differential  formulae  in 
case  it  is  thought  desirable.  See  Articles  128-131. 


§  124.  TIME   BY  ALTITUDE   OF    THE   SUN.  219 

Colorado,  1878,  July  28.9. 

Mean  solar  chronometer.  Observer  B. 

Negus  1326. 

Thermometer  78°. 

Latitude  q>     —  38°    4'    o"  Barometer        26.05 

Longitude  L  =          ih  44™  41"  w.  of  Washington. 
Assumed  A  T  =     —  6    41        7  INDEX  CORRECTION. 

On  Arc.  Off  Arc. 

31'  50"         359°  28'  45" 
31    30  28   40 

31   40  28   40 


31'  40"         359°  28'  42" 
Index  correction  =  I  =  —  n" 

From  the  refraction  table  we  find  Mean  refraction    =59".! 

Barometer  factor  =  .880 
Thermometer        =  .946 

Therefore  r  =  49".  2 
From  the  American  Ephemeris  we  find  — 

p.  248,  eq.  hor.  parallax  it  =  8".  72 

p.  327,  $  =  +  i8°42'  i6".7 

p.  327,  equation  of  time  £  =  -j-          6m  I28  .99 

p.  327,  semidiameter        s  =•  15'  47".  7 

d  is  interpolated  from  the  ephemeris  by  the  method  explained  in  Art.  52. 

The  ephemeris  is  given  for  the  meridian  of  Washington;  therefore  we  require 
the  Washington  time  of  our  observation. 

Time  of  observation  T  —         3h  37™  26".  3 

Approximate  correction     A  T  =  —    6    41      7 
Approximate  local  time  =        20    56    19 

Longitude  =          i    44    41 

Washington  time,  July  28         =       22    41      o 

_  jh  jgm  os  before  noon  of  July  29 


At  noon,  July  29,  d  =  18°  41'  29".  6 

Hourly  change  July  28  =  —  35".  oo 
Hourly  change  July  29  =  —  35".  77 
Therefore  the  correction  to  d  =  —  ih.3i7[—  35-77-H-77Xd.O55] 

+  47"-  i 
At  time  of  observation  d  —  18°  42'  i6".7 

At  noon  July  29.  eq.  of  time  =      +  6m  12".  89 

Correction  for  d.  055  =  .10 

£  =  6m  12".  99 

In  taking  E  from  the  ephemeris,  second  differences  need  not  be  considered  for 
this  purpose,  though  it  has  been  done  in  this  case. 


220  PRACTICAL   ASTRONOMY.  §  125. 

If  a  sidereal  chronometer  had  been  used  we  should  have  had 
only  to  convert  the  mean  time  t  -\-  E  into  sidereal  time,  when 
we  should  have  had  A  T  by  comparing  with  the  observed  time 
as  now.  It  may  be  remarked  also  that  in  using  a  sidereal 
chronometer  the  observed  sidereal  time  must  be  converted 
into  mean  solar  time  for  the  purpose  of  taking  d  and  E  from 
the  ephemeris,  since  these  are  given'  for  mean  solar  time. 

In  reducing  such  a  series  as  this  it  is  perhaps  a  little  better 
to  reduce  the  readings  on  the  two  limbs  separately;  the  two 
reductions  will  then  mutually  check  each  other.  Of  course 
the  altitudes  must  be  corrected  for  semidiameter.  If  a  con- 
siderable number  of  series  have  been  reduced  in  this  way 
the  observer  can  see,  by  comparing  results,  whether  his  per- 
sonal equation  is  the  same  for  both  limbs. 

125.  By  a  single  altitude  of  a  star. 

It  will  be  convenient  to  use  a  sidereal  chronometer  when 
practicable. 

Let  &  =  the  true  sidereal  time  of  observation  ; 

<90  =  the  chronometer  time  of  observation  ; 
=  the  chronometer  correction. 


Then  t  is  computed  the  same  as  above  ;  recollecting  that  for 
a  star  the  semidiameter  and  parallax  will  be  inappreciable, 
we  have 

,  =  go°-(*'-r);  ......   (219) 

@  =  (/  +  «)=  ©0  +  ^@; 

(/  +  a)  -  @0  .......     (220) 


§125- 


TIME  BY  ALTITUDE    OF  A    STAR, 


221 


Example  2.  West  Las  Aniraas,  Colorado.  1878,  July  29.3. 

Observation  of  Arcturus  for  time.  Observer  B. 

Sidereal  chronometer. 
Negus  1590. 


Sextant. 
87°  40' 
30 
20 
10 

87    oo 

Chronometer. 
IS1'  IIm  29s.  O 

ii    55  -o 

12      21   .0 
12      46  .5 
13      13  -0 

Latitude     cp  =  38°    4'  oo" 
Longitude  L  =  ih  44™  41*  w.  Of  Wash. 
Thermometer  74°.o 
Barometer        25  .91 

From  ephemeris,  a=i4h  iom  8s.  2 
6  =  19°  48'  58" 

A 

0.10386         -|-   9.9 
.02651 

9.72786             -f-  20.0 

938525              +50-5 

Means 

7 
£ 

87° 

20' 

oo" 

18 
42 

l8u  I2m  20s.  9 

sec  tp  = 
seed  = 

sin  6"  = 
sin  D  = 

2A  = 

A  - 
r  — 

h  = 
z  = 

6  = 

87° 
43 

43 
46 

38° 
19 

19' 

39 

38 

21 
4' 

oo" 

30 
46 

44 
16 

o" 

58 

f\ 

S  = 
D  = 

18° 
64 
28 
32 
14 

15' 
36 

6 
18 
3 

2" 
18 
14 
9 

7 

*,= 

24° 

44' 

33"-  3 

sin2  %t  = 
sin  \t  = 

9.24348 
9.62174 

+  27-5 

t  =  49    29     7 
t=    3"  I7-56-.5 
a  =  14    10     8  .2 
6=17    28     4.7  =  sidereal  time 
Observed  90=  18    12    20.9  —  chron.  reading 

40  =  —    44"'i6s.2  =  chron.  cor.  [Eq.  (220)]. 

It  will  be  seen  that  the  numerical  work  is  somewhat  less 
in  case  of  a  star  than  of  the  sun. 

In  case  a  mean  solar  chronometer  has  been  used,  the  side- 
real time  (t  -]-  a)  must  be  converted  into  mean  solar  time  by 
(202),  and  the  resulting  value  compared  with  the  chronome- 
ter time. 


222  PRACTICAL   ASTRONOMY.  §  126. 

Example  3.  West  Las  Animas,  Colorado.  1878,  July  27.3. 

Observation  of  a  Coronce  Borealis  for  time.  Observer  B. 

Mean  solar  chronometer. 
Negus  1326. 


Means 
E 

Sextant. 
95°  50' 
40 
30 
?o 
95    10 

95°  30'    o" 
o 
-  52 

Chronometer. 
I7h  3:n  i6s.o       Latitude     ^>  =  38°    4'  oo" 
3    40.0       Longitude  L=    ih44m  4i8w.ofWash. 
4      5  .0       Thermometer  62°.  o 
4    32.5       Barometer        26.11 
17    4    57-5 

I7h  4m    6s.  2    From  the  ephemeris,  <x=i5h29m34M 
5=27°  7'  32" 

A* 
sec  (p  =  0.10386          -j-    9.9 
sec  d  =    .05061          +4-5 

sin  S  =  9.65113          -{-25.2 
sin  D  =  9.43137          -T  45-0 

2A  = 

A  - 

95    29     8 
47    44  34 
-46 

h  - 
z  ^= 

?  = 

cp  —  d  = 

I  -  (%>-  d)  = 
D  = 

47°  43'  48" 
42    16   12 

38°    4'    o" 
27      7  32 
10    56  28 
53    12  40 
31    19  44 
26    36  20 
15    39  52 

t  = 

<T  = 

e  = 

v  = 
e  —  F  = 

M.  S.  time  = 
Chronom.  = 

24°  32'  43" 
49      5   26 
3h  i6m2i8.7 
15    29  34.1 

18    45   55.8  = 

8    21    15.7  = 
10    24  40  .  i 
i  42.3 

10     22    57.8 

17      4     6.2 

6   41     8  .4 

sin2  \t  =  9.23697 
sin  \t  =  9.61848          +27.7 

sidereal  time.     This  is  now  converted  into  mean 
solar  time  by  equation  (202). 
sidereal  time  of  mean  noon  from  ephemeris 

Table  II,  Appendix  to  Ephemeris. 

126.  Conditions  most  favorable  to  accuracy  in  determining  time  by  a  single 
altitude. 

As  our  data  will  always  be  liable  to  more  or  less  uncertainty,  it  becomes  a 
matter  of  great  practical  importance  to  so  arrange  our  observations  that  small 
errors  in  the  quantities  regarded  as  known  shall  have  the  least  effect  on  the 
computed  value  of  /. 

*  These  quantities  are  written  down  so  that  we  may  employ  them  in  computing  the  differen- 
tial formulae  when  desirable.  (See  Articles  128-131.) 


§  128.  DIFFERENTIAL   FORMULAE.  22$ 

As  we  require  equations  (121),  we  rewrite  them  here  for  convenience  of  ref- 
erence. 

cos  h  cos  a  =  cos  6  cos  /  sin  <p  —  sin  6"  cog  <p;       (<?)  } 

cos  h  sin  a  —  cos  6"  sin  /;  (f)  >•     .     .     (121) 

sin  h  =  cos  d  cos  /  cos  cp  -\-  sin  6"  sin  (p.     (g)  } 

To  determine  the  effect  upon  t  of  a  small  error  in  the  measured  altitude.  Differ- 
entiating (g)  with  respect  to  h  and  t  and  reducing  by  means  of  (f),  we  readily 
find 

dt  = —. dh (221) 

cos  <p  sin  a 

From  this  we  see  that  for  a  given  latitude  cp  a  small  error  dh  in  the  altitude 
will  produce  the  least  effect  when  sin  a  has  its  greatest  value,  viz.,  when  the  star 
is  on  the  prime  vertical.  Also,  that  for  a  constant  positive  error  dh  the  error 

produced  in  /  will  be   T    when  the  star  is  j  r  of  the  meridian,  and  may 

therefore  be  eliminated  by  observing  both  east  and  west  stars. 

(221)  also  shows  that  dt  will  be  least  when  cos  q>  is  greatest,  that  is,  when 
<p  is  small;  the  most  favorable  part  of  the  earth's  surface  for  this  kind  of  deter- 
mination being  the  equator. 

Effect  of  a  small  error  in  the  assumed  latitude  (p.  Differentiating  (g)  with  re- 
spect to  q>  and  t  and  reducing  by  means  of  (e)  and  (/),  we  find 

dt  = dq>\ (222) 

tan  a  cos  <p 

from  which  it  appears  that  when  the  star  is  near  the  prime  vertical  dt  is  rela- 
tively small.  If  the  star  is  on  the  prime  vertical,  dt  is  zero,  as  tan  a  is  then 
infinite. 

If  the  star  is  not  observed  on  the  prime  vertical,  dt  will  disappear  from  the 
mean  of  two  observations  at  the  same  distance  east  and  west  of  the  meridian. 
Also,  we  see  that  an  error  dq>  will  have  the  least  effect  on  /  when  the  latitude  is 
near  zero. 

In  the  same  way  we  may  discuss  the  effect  of  a  small  error  in  6";  but  as  no 
stars  will  ever  be  likely  to  be  used  for  this  purpose  whose  declination  is  uncer 
tain  to  any  appreciable  amount,  this  is  not  practically  a  source  of  error. 

127.  From  this  discussion  we  see  that  a  determination  of  time  should  always 
depend  on  observations  of  stars  both  east  and  west  of  the  meridian;  the  obser- 
vations should  be  made  at  as  nearly  the  same  azimuth  as  possible  east  and  west, 
and  if  two  stars  are  employed  it  will  be  better  if  the  declinations  are  nearly  equal. 

dh  may  be  regarded  as  including  all  of  the  undetermined  errors  of  the  instru- 
ment— see  Articles  115,  116,  and  117 — as  well  as  constant  errors  of  observation 

and  refraction. 

Differential  Formula, 

128.  The  numerical  values  of  the  differential  coefficients  of  ^with  respect  to 
(p,  d,  and  zh  are  often  convenient  where  the  time  has  been  determined  in  tfie 


224  PRACTICAL   ASTRONOMY.  §  129. 

manner  just  explained.  Sometimes  values  of  cp,  d,  or  2/1  are  employed  in  the 
computation  which  are  afterwards  found  to  require  small  corrections.  If  these 
are  so  small  that  the  second  and  higher  powers  may  be  neglected,  the  necessary 
correction  of  the  hour-angle  may  be  found  by  the  differential  formula.  Other- 
wise the  computation  must  be  repeated. 

Let  Acp,  AS,  Alh  =  small  corrections  to  the  values  of  the  latitude,  declina- 

tion, and  double  altitude  employed; 
At  =  the  resulting  correction  to  the  hour-angle. 
Then,  neglecting  terms  of  the  second  and  higher  orders, 


The  differential  coefficients  may  be  computed  by  the  formulae  of  the  previous 
article,  but  they  are  not  convenient  since  they  require  a  knowledge  of  the  azi- 
muth. 

129.  For  practical  purposes  a  more  convenient  process  is  the  following, 
where  the  numerical  values  of  these  coefficients  are  expressed  in  terms  of  the 
differences  of  the  logarithms  employed.  Taking  logarithms  of  both  members 
of  (215),  we  have 

2  log  sin  \t  =  log  sin  S  -\-  log  sin  D  -J-  log  sec  cp  -\-  log  sec  <5;      .     (224) 
where  S  =  i|>  -f  (<p  -  S)J  =  |9o°  -  &A  +  i(<p  -  5);  )  .      } 

Z>  =  i|>  _  (0,  -  d)]  =  |9o°  -  *2A  -  i(<?  -  d).  } 
First  differentiate  (224)  with  respect  to  2k  and  %t.     We  find 

2dl  sin  \t  _  dl  sin  S     dS_     dih       dl  sin  D     (W    dzh 
~  '  d±t  dS      '  d2h  '  ~d\t  ~r       dD       '  d2h  '  dtf 

dS        dD  i 

From  (225),  ^  -  ^  -  -  -. 

dl  sin  ±t  ,  ^//sin  S         ...      „ 

Therefore  we  nave,  writing  —  ^7-  =  Z//sm  \t  and  -  ds      =  A  Ism  S.  .  .  , 


dt    _          Al  sin  S  +  ^?/sin  D 
Hhh  ~  4  Al  sin  \t 

The  quantities  Al  sin  S,  Al  sin  D  .  .  .  are  the  rates  of  change  of  the  loga- 
rithms for  the  values  of  S,  D,  etc.,  employed.  It  requires,  therefore,  very  little 
time  to  take  these  from  the  tables  while  computing  t,  as  we  have  done  in  the 
examples  in  the  foregoing  pages. 

Thus,  in  example  i  we  have  found  Alsin  S=  19  9,  which  is  the  change  expressed 
in  units  of  the  last  decimal  place  of  log  sin  S  produced  by  a  change  of  i'  in  S. 
In  practice  the  /  sin  of  the  angle  5'  less  than  S  is  subtracted  from  that  of  the 
angle  5'  greater,  and  the  difference  divided  by  10.  This  is  a  little  more  accu- 
rate than  to  take  the  difference  between  consecutive  logarithms. 


§131.  DIFFERENTIAL   FORMULA.  22$ 

In  our  example  S  =  32°  24' 

/sin  32°  19'  =  9.72803 
/sin  32°  29'  =  9.73002 

Difference  for  10'  =          199 
Difference  for    i'  =A  =  ig.g 

In  like  manner  we  have  found          Als\nD  =        54.6 

A  I  sin.  i/  =  —  28.7 


Therefore,by(226),  =- 


A  correction  to  the  assumed  value  of  2/1  may  result  from*a  variety  of  causes, 
such  as  the  employment  of  values  of  the  refraction,  parallax,  index  error,  or 
eccentricity,  which  are  only  approximately  correct,  or  from  errors  in  the  pre- 
liminary computation. 

Suppose  the  value  of  2,h  employed  in  example  I  was  found  to  require  the  cor- 
rection Azh  —  i'.  Then  the  resulting  correction  to  the  hour-angle  would  be 

60" 
At  =  .649  X  ---  =  2".  596. 

130.  For  the  value  of  —^  we  differentiate  (224)  with  respect  to  t  and  d,  viz., 

zdl  sin  $t  _  dl  sec  £      d8_       dl  sin  S    dS     dd        dl  sin  D    dD      d$ 
~~d\t  dS       '  7\t  "T       dS      '  7<5  '  1#  ~r  ~~d~D       '  ~d8   '  ~d\? 

,      .  dS  i  dD         .    i 

and  from  (225),  ^  =  -  -;  -^  =  +  -. 


„,        .  dt       2  Al  sec  d  -  Al  sin  S  +  Al  sin  D 

Therefore  -^  =  ^-^T^Ti^  --  .....     (227> 

Substituting  the  numerical  values  of  J/sec  S,  J/sin  6",  etc.,  given  in  exam- 
ple i,  we  find 

dt  _  8.6  -  19.9  4-  54.6 
^  =  -  57-4  ~  '754' 

If  now,  for  example,  the  S  with  which  the  reduction  is  made  were  found  to 
require  the  correction  Ad  =  i',  we  should  have 


131.  For  —  —  we  differentiate  with  respect  to  <p  and  t.  viz., 
d  <p 

2dl  sin  \t  _  <//sin  S     dS      d<p       dl  sin  D  dD      dtp       dl  sec  <p     dcp 
d\t  dS     '~d^'~d\t^  ~  dD      d     '  ~dt~\        ~dl>       '7i' 


226 


PRACTICAL   ASTRONOMY. 


§132. 


Also, 


Therefore 


ds  _  x.  -£  —     - 

dq>  ~  2'  ~d(p  2' 

dt_  _  iAl  sec  <p  -f  J/  sin  S  — 


sin  D 


sn 


(228) 


For  our  example  I  we  have  by  this  formula 

dt_  _  19.8  4-  19.9  -  54.6  _  _    26o 

d<?  57-4 

and  a  correction  of  i    to  the  assumed  latitude  produces  a  corresponding  cor- 

rection to  the  time  of 

60" 

At  =  —  .260  —  =  —  i8.  04. 
4 

Probable  Error. 

132.  By  means  of  formula  (226)  we  may  reduce  the  time  of  each  altitude  to 
the  time  of  the  mean  altitude  for  the  purpose  of  comparing  the  individual  meas- 
urements and  computing  the  probable  error.  The  application  to  example  i 
will  sufficiently  explain  the  process. 

The  mean  value  of  2h  is  89°  10',  so  that  each  time  will  be  reduced  to  the  time 
corresponding  to  this  altitude.  Further,  as  one  half  the  readings  were  made  on 
the  lower  limb  and  one  half  on  the  upper  limb,  we  must  add  to  the  latter  and 
subtract  from  the  former  the  time  required  for  the  sun  to  move  in  altitude  over 
an  arc  equal  to  the  sun's  semidiameter,  or  in  double  altitude  a  space  equal  to 
the  diameter. 

Thus  we  have  —  see  example  i  — 

Semidiameter  of  sun  =  S  =  15'  47".  7; 
Diameter  of  sun          =  31'.  590. 
dt 


From  previous  article, 

Therefore  reduction  for  semidiameter 
The  reduction  is  now  as  follows: 


d2h 


=  .649. 

=  .649  X 


31'. 590  X  60 

15 


=  828.oi. 


Limb. 

Observed 

2*. 

*i 

A*. 

Correction 
for  Semi- 
diameter. 

Observed 
Time. 

Reduced 
Time. 

V. 
0  —  C. 

"V  V. 

Upper 

88-  50' 
89      o 

+  20' 
+  10 

+  26  '.o 

+  Im  22«.0 

3h  35m  12g  0 
35     39  -5 

3h  37m  25*-9 
27  -5 

+.:l 

16 
144 

89    10 

O 

.0 

36       3    5 

-    .8 

64 

89      20 

—    10 

-26.0 

36     30  -5 

26  .5 

+      -2 

4 

89      30 

—  20 

—  51  .9 

36     56  -5 

26  .6 

+    -3 

9 

Lower 

88    50 

+  20 

+  5i  -9 

—  I  22  .0 

3    37     55  -5 

25-4 

—    -9 

81 

89      o 

+  10 

+  26.0 

38      22  . 

26  .0 

—    .3 

9 

89    10 

O 

.0 

38      48. 

26  .0 

—    .3 

9 

89     20 

—    IO 

-26.0 

39     H  -5 

26.5 

+      -2 

4 

89     30 

—  20 

—  51  .9 

39    4i  -o 

3    37     27.1 

+    -8 

64 

Mean 


=  4-04 


§  134-  DIFFERENTIAL  FORMULAE.  22  7 

Then  by  formulae  (27),  probable  error  of  single  observation  =  r  =  '.43; 
probable  error  of  mean  =  r0  =  8.i4. 

The  reader  must  not  fall  into  the  error  of  supposing  that  this  quantity  repre- 
sents the  actual  probable  error  of  a  determination  of  time  by  this  method,  since 
no  account  is  here  taken  of  the  relatively  large  constant  errors  to  which  observa- 
tions of  this  kind  are  liable.  The  subject  will  be  considered  more  at  length 
hereafter.  (See  Art.  156.) 

Corrections  for  Refraction  and  Motion  in  Declination. 

133.  The  refraction  of  the  atmosphere  and  the  sun's  motion  in  declination 
affect  the  computed  value  of  At  by  small  quantities,  which  it  may  be  considered 
desirable  to  take  into  account  in  a  more  refined  discussion. 

Correction  for  Refraction.  Since  refraction  decreases  with  the  altitude,  it  fol- 
lows that  when  the  sun's  altitude  increases  by  a  given  quantity — 10'  for  example 
— as  measured  with  the  instrument,  the  actual  space  passed  over  is  greater  than 
10'  by  the  difference  of  refraction  for  the  first  and  last  position.  Thus,  instead 
of  simply  Azh  as  used  in  our  formula,  we  should  employ  Azh  -\-  zAr,  Ar  be- 
ing the  difference  between  the  refraction  for  altitude  h  and  that  for  h  -f-  Ah. 

For  our  example  we  find  for  the  mean  altitude  of  the  sun,  viz.,  44°  34', 

Change  in  refraction  corresponding  to  10'  altitude  =  o".3O  =  zAr. 
Therefore  the  correction  to  At  corresponding  to  Aih  =  10'  is 

.649  X  •—  =      '.013 

This  must  be  added  to  the  computed  interval,  viz.,  At  =  25^.96 

A't  =  258.973 

134.  Correction  far  Sun's  Motion  in  Declination.  Since  the  sun's  declination  is 
not  constant,  but  is  ever  increasing  or  diminishing,  the  time  required  for  the 
altitude  to  change  by  a  given  amount  will  be  slightly  modified  by  this  cause. 

For  our  example  with  Aih  =  10'  we  find  At  =  25". 97.  Referring  to  the 
example,  we  have  found  the  hourly  motion  in  declination  to  be  —  35". 7;  there- 
fore in  the  interval  25s. 97  the  change  is  —  ".26. 

By  formula  (227)  we  have  found  for  this  example  — ^  =  —  .754. 

do 

"  26 

Therefore  correction  to  At  —  —  .754  X  -  =  +  ".013. 

Therefore  the  final  value  of  At  corresponding  to  Azh  =  10'  is  259.986. 


228  PRACTICAL   ASTRONOMY.  §  'US 

If  both  limbs  are  reduced  together,  as  in  our  example,  the  reduction  for  semi- 
diameter  should  be  corrected 'for  motion  in  declination,  but  not  for  refraction 
since  both  limbs  are  observed  at  the  same  altitude. 


Determination  of  Time  by  Equal  Altitudes. 

135.  By  a  star  observed  at  equal  altitudes  east  and  west  of  the 
meridian. 

Method  of  observing.  When  the  star  is  at  some  distance 
east  of  the  meridian  (the  nearer  the  prime  vertical  the 
better),  measure  with  the  sextant  a  series  of  five  or  more 
altitudes  in  the  manner  already  explained  (Arts,  in,  112, 
and  113);  then,  a  short  time  before  the  star  reaches  the 
same  altitude  in  the  west,  set  the  vernier  at  the  reading 
of  the  last  altitude  and  observe  the  same  number  of  alti 
tudes  as  before  at  the  same  readings.  Some  observers 
prefer  to  take  only  one  reading  east  and  then  lay  the  in- 
strument where  nothing  will  disturb  it  until  it  is  time  for 
the  west  observation.  In  this  way  both  observations  are 
secured  at  absolutely  the  same  altitude  so  far  as  it  depends 
on  the  reading  of  the  instrument ;  but  there  is  the  objection 
that  only  one  reading  can  be  made,  which  more  than  neutral- 
izes the  advantage.  No  correction  for  index  error,  refrac- 
tion, or  parallax  is  required. 

Now,  as  the  declination  is  constant  and  the  altitudes  the 
same,  the  numerical  values  of  the  hour-angle  measured  east 
and  west  of  the  meridian  will  be  equal.  Suppose  a  sidereal 
chronometer  used.  Let 

0'    =  the  chronometer  time  of  the  first  observation; 
©"  =  the  chronometer  time  of  the  second  observation; 
AS    =  the  chronometer  correction. 

Then  the  sidereal  time  of  the  star's  meridian  passage  equals 
its  right  ascension  a. 


§  136.  EQUAL  ALTITUDES  OF  A    STAR.  22$ 


For  the  first  observation       a  =  @'  -|-  J<9  -f-  t\ 
For  the  second  observation  a  =  ®"-\-  A®  —  t. 

From  which  J®  =  a  -  £(©'  +  ©")  .....     (229) 

Example  i.     1856,  March  igth,  equal  altitudes  of  Arcturus 
east  and  west  of  the  meridian  were  observed  as  follows  : 

East  of  meridian,   &   —  nh    4m  5is.5 
West  of  meridian,  @"  =  17   21    30.0 


=  14   13    io.75 
From  ephemeris,      a  =  14     9      7.11 


Therefore  J©  =     —  4m    38.64 

136.  If  a  mean  time  chronometer  is  employed,  the  sidereal 
time  of  the  star's  culmination  (which  is  equal  to  the  right 
ascension)  must  be  converted  into  mean  time,  and  this  com- 
pared with  the  mean  of  the  observed  times  as  before. 

Example  2.  1856,  March  I5th,  equal  altitudes  of  Spica  were 
observed  as  below,  the  time  being  noted  by  a  mean  time 
chronometer: 

Latitude     cp  =  —  33°  56' 

Longitude  L  —  —  ih  13"  s68  from  Greenwich. 

CHRONOMETER.  SEXTANT.  CHRONOMETER. 

East.  Double  Alt.  West. 

I0h  20ra     0s.  5  104°     O'  2h  40""  38". 

20      28  10  40      10.5 

20    55  20  39    42 


T'  =  ioh  20™  278.83  T"  =  2h  40m  ioM7 

"}  =  12   30    19.0 

From  ephemeris,  a  =  ©  =        13    17     37  .92 
Then  —  Art.  95  —  from  ephemeris          V—       23   32     53.22 


9  -  V  =       13   44  44.70 

Table  II,  ephemeris,  —  2  15  .12 

Mean  time  =        13    42  29.58 

i(T'  -f  T")  =        12    30  19.00 

Therefore  A  T  =  -\-    i    12  10  .58 


230  PRACTICAL   ASTRONOMY.  §  137. 

137.  By  equal  altitudes  of  the  sun. 

This  method  is  less  simple  when  applied  to  the  sun,  for  the 
reason  that  the  sun's  declination  cannot  be  considered  con- 
stant for  the  interval  of  time  between  the  morning  and  after- 
noon observations.  The  mean  of  th&  observed  times  will  not 
therefore  be  the  time  of  meridian  passage  as  in  case  of  a  star. 
The  correction  due  to  this  cause  is  called  the  equation  of  equal 
altitudes.  To  determine  its  value  we  proceed  as  follows : 

Let  Ad  =  the  hourly  change  in  declination  taken  from  the 

Nautical  Almanac. 
Then  tAd  =  the  total  change  in  d  in  the  time  / ; 

dt  =  change  produced  in  /  by  the  increment  tAd  of  #. 

Then  since  t  = 


and  neglecting  terms  of  higher  order  than  the  first, 

dt 


(230) 


To  determine  ^  we  differentiate  the  last  of  equations  (121) 
with  respect  to  /  and  tf,  viz., 

dt    _  sin  cp  cos  d  —  cos  cp  sin  8  cos  t  _  tan  <p      tan  d 
dd  ~  cos  q>  cos  d  sin  t  ~  sin  /       tan  /  ' 

Therefore  substituting  this  value  in  (230),  and  dividing  by 
15,  as  8t  is  required  in  seconds  of  time,  we  find 

tan  <T1    Ad 


Now  suppose  a  mean  time  chronometer  used,  and  let 

T"  =  chronometer  times  of  east  and  west  observation, 


§  138.  EQUAL  ALTITUDES   OF   THE   SUN.  231 

Then  will 

/  —  dt  =  the  hour-angle  of  the  A.M.  observation  ; 
t  -f-  dt  —  the  hour-angle  of  the  P.M.  observation  ; 
E  —  equation  of  time. 

Then     E  =  T'  +  A  T  +  (t  —  dt)  from  A.M.  observation  ; 
E  —  T"  +  4T  —  (t  +  df)  from  P.M.  observation. 

From  which 

T")-*f]  .....     (232) 


Example  3.  1856,  March  5th,  at  the  U.  S.  Naval  Academy 
the  sun  was  observed  east  and  west  of  the  meridian  as 
follows  : 

East,     T'  =  ih    8m  268.6 

West,  T"  =8   45    41  .7  Latitude  <p  =       38°  59' 

Longitude  L  =  —    2m  i68 


=  %(T"  —  T')  =  3h  48m  37B.5  from  Washington 

=  57°  9'  From  ephemeris,  d  =  —    5°  46' 

=  3b.8io  Equation  of  time  E  =  -|-  nm  35*. n 

AS  =  4-        *8".io 

}(^'+  r")-4h57m    4M5 
dt  =  +          15.18 

E  —  -\-  ii     35  .11  tan  <p  =  9.9081           tan  d  =  9.oo42n 
sin  t  —  9.9243            tan  /  =    .1900 


AT  —  —  4h45m  13".  86 


9.9838  8.8i42n 

*A  =  1.1696  *B  =  1.1980 

log  /  =    .5809 
log  AS  —  1.7642 
log^  =  8.8239 

log  8t  =  1.  1812 

138.  Equal  altitudes  of  the  sun  observed  in  the  afternoon  of 
one  day  and  the  morning  of  the  day  following. 

In  this  case  the  mean  of  the  observed  times  plus  the  neces- 

*  See  tables  of  addition  and  subtraction  logarithms. 


232  PRACTICAL  ASTRONOMY,  §  138. 

sary  corrections  will  be  the  time  of  the  sun's  passing  the 
lower  branch  of  the  meridian,  or  midnight. 

Let  t'  =  the  sun's  hour-angle,  reckoned  from  the  lower 
branch  of  the  meridian. 

Then  *'  =  *  +  180°  ;        sin  t  =  —  sin  tf  ;        tan  /  =  tan  tf. 
Therefore  for  this  case  (231)  becomes 

n  c       tan 


and  the  clock  correction  will  be  given  by  (232),  as  before, 
except  that  for  E  we  write  I2h  -f-  E. 

Example  4.  1856,  May  3d.  The  altitude  of  the  sun  being 
observed  on  the  afternoon  of  the  3d  and  the  morning  of  the 
4th  as  follows,  required  the  correction  of  the  chronometer 
at  midnight. 

T1  =    6h  54m  io8.3  Latitude  south  =  <p  =  —  43°  21' 

T"  =21     917  -5        Longitude  W.  of  Wash.  =  L  =  -j-  gh  ira  40* 


"  —  T')  =  /'  =    7h    7m  34V  From  ephemeris,  S  =       15°  15' 

t'  =106°  53'  JS  =  +43".76 

/'  =    7h.i26  Equation  of  time  E  =  —  3m  i88.67 


i(T"  +  T1)  =  I4h    im439-9  tan  <P  =  9-9750»  tan  S  =  9.4356 

dt  =  22.2  sin  tr  =  9.9809  tan  /'  =    .5I79» 

I2h  +  ^  =  II    56    41  .33 


AT  '=  —    2h   4m408.4  A  =  1.0764  B  —  1.1114 

log  t  —    .8528 
log  AS  =  1.6411 
log  TV  =  8.8239 
log  (—  6V)  =  1.3469^ 


§  140.  LATITUDE,  233 

139.  The  chief  advantages  possessed  by  the  method  oi 
determining  time  by  equal  altitudes  are  the  following:  the 
computation  is  very  simple,  and  no  corrections  are  required 
for  parallax,  refraction,  semidiameter,  or  instrumental  errors, 
nor  is  a  knowledge  of  the  latitude  required,  except  very  rough- 
ly, when  the  sun  is  employed.  The  disadvantages  are  the  di  ffi- 
culty  and  often  impossibility  of  obtaining  the  observations  at 
exactly  the  same  altitude,  owing  to  clouds  or  other  hinder- 
ances ;  also,  the  changes  which  often  take  place  in  the  re- 
fraction between  the  morning  and  afternoon.  A  correction 
for  this  last  mentioned  source  of  error  may  be  computed  by 
means  of  a  differential  formula,  but  it  has  not  been  thought 
necessary  to  develop  it  here. 


Latitude. 

140.  We  have  seen  (Art.  63)  that  the  astronomical  latitude 
of  any  place  is  equal  to  the  declination  of  the  zenith  of  that 
place,  or  to  the  elevation  of  the  pole  above  the  horizon.     The 
distinctions  between  the  different  kinds  of  latitude,  as  defined 
in  Art.  73,  must  be  borne  in  mind.     We  are  at  present  only 
dealing  with  the  astronomical  latitude  as  there  defined.     It  is 
perhaps  unnecessary  to  state  -that  all  formulas  derived  will 
be  applicable  to  either  north  or  south  latitude,  care  being 

taken  to  use  the  proper  algebraic  signs: 
and  declinations  being  j  ^^s. 

First  Method. 

141.  By  the  zenith  distance  of  a  star  observed  on  the  meridian. 
Resuming  the  last  of  equations  (121), 


2 34  PRACTICAL  ASTRONOMY.  §  141. 

cos  z  =  sin  cp  sin  d  -[-  cos  cp  cos  d  cos  /, 
we  know  that  when  the  star  is  on  the  meridian, 

t  =  o;          cos  t  —  i. 
Therefore  we  have 

cos  z  =  cos  (<p  —  tf)  ; 
±  #  =  <p  —  d    and     <p  =  tf  ±  <s-.      .     .     (234) 

By  referring  to  the  figure,  ES  =  #,  zS  =  z,  and  we  readily 
see    that  in  the  above  formula  the  sign  will  be  ±  for  a 


The  same  formula  applies  to  a  star  S"  observed  below  the 
pole.  If  we  reckon  the  declination  on  that  branch  of  the  me- 
ridian which  contains  the  observer's  zenith,  or,  what  is  the 


§142.  LATITUDE.  235 

same  thing,  if  we  replace  8  in  formula  (234)  by  (180°  —  #),  it 
then  becomes 

9  =  (180°  -  *)  -  * (235) 


Second  Method. 

142.  By  a  circumpolar  star  observed  at  both  upper  and  lower 
culmination. 

From  (234)  we  have— 

For  upper  culmination  cp  =  $  —  z ; 

For  lower  culmination  cp  =  180°  —  $  —  z1. 

The  mean  of  which  gives  cp  =   .90°  —  \(z  -\-  z'\    (236) 

The  method  has  this  advantage,  viz.,  that  the  latitude 
determined  in  this  way  does  not  require  a  knowledge  of  the 
place  of  the  star;  it  is  therefore  especially  adapted  to  the 
determination  of  the  latitude  of  a  fixed  observatory,  where  it 
is  desirable  to  make  the  results  independent  of  what  has  been 
done  at  other  places.  As  will  appear  hereafter,  when  extreme 
accuracy  is  required  there  will  be  a  small  correction  neces- 
sary for  the  change  in  d  between  the  first  and  second  observa- 
tion. The  result  is  also  affected  by  whatever  error  there 
may  be  in  the  tabular  value  of  the  refraction  used. 

The  following  example  will  illustrate  both  the  above 
methods : 

1875,  November  nth,  at  the  Washington  observatory  the 
zenith  distance  of  Polaris  was  observed  as  follows : 

Upper  culmination  z  =  49°  45'  22". 2  ; 
Lower  culmination  z'  =  52°  27'  2o".o . 


236  PRACTICAL   ASTRONOMY.  §  143. 

From  the  Nautical  Almanac  we  find  for  the  declination  of 
Polaris  at  the  time  of  upper  culmination  at  Washington : 

Nov.  ii. 4,  6  =  88°  39'    2". 8 
z  =  49   45    22  .2 


Therefore,  formula  (234),  <p  =  S  —  z  =  38°  53'  40".  6 

Also  for  lower  culmination,      Nov.  11.9,  £  =  88    39      3  .o 

Z1   =  52     27     20   .0 


Then  formula  (236)  gives  (p  =  180°—  d  —  z'  =  38°  53'  37".  o 
The  mean  of  these  values  gives  us  cp  =  38°  53'  38".  8 
By  the  second  method  we  have 

cp  =  90°  -  i(z  +  *>)  =  38°  53'  38".  9 

Third  Method. 

143.  By  an  altitude  of  a  star  observed  in  any  position,  the  time 
being  known. 

©,  the  sidereal  time,  is  known  ;  «,  the  right  ascension,  and 
6,  the  declination,  are  taken  from  the  Nautical  Almanac. 

We  then  have  t  =  &  —  a. 

This  will  be  given  in  time,  and  must  be  multiplied  by  15 
to  reduce  it  to  arc.     We  then  have 

sin  h  =  sin  cp  sin  d  -f-  cos  (p  cos  tf  cos  /  ; 

in  which  q>  is  the  only  unknown  quantity. 

For  solving  the  equation  introduce  two  auxiliaries,  <a?and  D, 
determined  by  the  equations 

d  sin  D  —  sin  d  ;     .......      (a) 

d  cos  D  =  cos  6  cos  /  ......     (a') 

The  above  equation  then  becomes,  by  substituting  the  value 
of  d  from  (a\ 

cos  ((p  —  D)  —  sin  h  sin  D  cosec  $. 


§  143-      ALTITUDE   OBSERVED  AT  ANY  HOUR-ANGLE. 

Dividing  (a)  by  (a'}  to  determine  D,  we  have  the  following 
formulae  for  determining  cp : 

tan  D  —  tan  d  sec  / ;  ) 

cos  (<p  —  D)  —  sin  h  sin  D  cosec  d.  } 

D  is  taken  less  than  90°,  +  or  —  according  to  the  algebraic 
sign  of  the  tangent.  (y>  —  D),  being  determined  in  terms  of 
the  cosine,  may  be  either  +  or  — .  There  will  therefore  be 
two  values  of  the  latitude  which  will  satisfy  the  above  condi- 
tions. Practically  an  approximate  value  of  the  latitude  will 
always  be  known  with  accuracy  sufficient  for  deciding  this 
ambiguity. 

Example.  On  March  4th,  1882,  I  observed  the  following 
double  altitudes  of  Polaris  with  a  Pistor  &  Martins  prismatic 
sextant  and  artificial  horizon  : 


Sextant. 

Clock. 

79°  12'    o" 

I0h  43m    48 

10  50 

43    56 

10  30 

45      2 

10     5 

45    50 

9  50 

47    45 

Means     79°  10'  39" 
Index  correction  /    —  i     2.0 

ioh45m    78-4 
A®           +  i  .5 

2h'   = 

79°    9'  37" 

0  =  ioh  45m    8s.  9 

h1   = 

Refraction 

39    34  48.5 
-     i     9-7 

a  =    i    15      6  .0 

*  = 

39    33  38.8 

t=  142°  30'.  43".  5 

8  =    88°  41' 
t  —  142    30 

6".  2 

43  -5 

tan  =  1.6391390 
cos  =  9.8995369* 

From  Nautical  Almanac  : 

ih  i5m  6."o 
=  88°  41'  6".2 


cosec  S  =  .0001144 


D  —  —  88  57  23  .6  tan  D  =  1.7396021*       sin  D  —  9.9999279* 

h  =  39  33  38  .8                          Sin  h  =  9.8040688 

<?  —  D  =  I29  33  55  -4                     cos  (cp  —  D)  =  9.8041111* 

<p  =  40  36  31  .8 


238  PRACTICAL  ASTRONOMY.  §  H5- 

In  this  example  there  is  no  ambiguity  :  cos  (9  —  D)  being 
negative,  the  angle  must  be  in  the  second  or  third  quadrant. 
If  we  had  taken  it  in  the  third  quadrant  we  should  have 
found  cp  —  141°  +.  As  <p  is  never  greater  than  90°,  this  value 
is  in  any  case  excluded. 

144.  Effect  of  Errors  in  the  Data  upon  the  Latitude  determined  by  an  Altitude 
of  a  Star. 

Differentiating  equation  (£•),  Art.  126,  regarding  h  and  q>  as  variable,  and 
reducing  by  equation  (e),  we  readily  find 


From  this  we  see  that  a  small  error  in  the  measured  altitude  will  have  the  least 
effect  on  the  latitude  when  the  star  is  on  the  meridian. 

Again,  differentiating  the  same  equation  with  respect  to  q>  and  /,  and  reduc- 
ing, we  readily  find 

d<p  =  —  tan  a  cos  tpdt  ;      .......     (239) 

from  which  it  appears  that  the  effect  upon  <p  of  a  small  error,  dt,  in  the  hour. 
angle  will  be  least  when  a  is  zero  or  180°. 

It  appears,  therefore,  that  the  latitude  will  be  determined  with  greater  accu- 
racy the  nearer  the  star  is  to  the  meridian.  When  the  star  is  very  near  the 
meridian  the  method  which  follows  will  be  preferable. 

Fourth  Method. 

145.  By  cir  cummer  idian  altitudes.  When  the  latitude  is 
determined  by  the  altitude  of  a  star  observed  on  the  meridian, 
the  accuracy  is  greater  than  in  any  other  position,  and  at  the 
same  time  the  computation  is  extremely  simple.  We  can, 
however,  only  measure  one  altitude  when  the  star  is  on  the 
meridian;  and  frequently  at  the  time  when  the  observation  is 
made  we  shall  not  know  the  chronometer  correction  with 
sufficient  accuracy  for  determining  the  exact  instant  when 
this  observation  should  be  taken.  If,  however,  altitudes  are 
measured  near  the  meridian  (how  near  we  shall  discuss  later), 
the  observed  altitudes  may  be  reduced  to  the  meridian  alti- 
tude by  a  simple  computation.  It  will  thus  be  possible  to 


§  145-     LATITUDE  BY  CIRCUMMERID1AN  ALTITUDES.         239 

make  a  considerable  number  of  measurements  instead  of  rely- 
ing on  one  alone.  When  this  method  is  applied  observation 
is  begun  if  possible  a  few  minutes  before  culmination,  and  a 
series  of  altitudes  measured  in  quick  succession  so  as  to  have 
about  the  same  number  on  each  side  of  the  meridian. 

Altitudes  measured  in  this  manner  are  called  circumme- 
ridian  altitudes. 

It  is  not  essential,  however,  that  the  series  should  be 
symmetrical  with  respect  to  the  meridian  ;  the  method  is 
equally  applicable  to  the  reduction  of  one  or  more  altitudes 
taken  on  either  side  of  the  meridian  if  sufficiently  near. 

Let  h  =  any  altitude  of  a  star  corresponding  to  the  hour- 
angle  / ; 

h0  =  the  altitude  when  the  star  is  on  the  meridian ; 
ZQ  =  the  zenith  distance  =  90°  —  kQ  =  <p  —  d 
Then 

sin  h  =  sin  cp  sin  d  -|-  cos  g>  cos  #  cos  t. 

Let  us  write  for  cos  t  its  value,  1—2  sin2  £/. 
Then  the  above  equation  becomes 

sin  h  =  cos  z  =  cos  (<p  —  6)  -    cos  cp  cos  d  2  sin'-J/.    (a) 
Let  us  write         cos  (p  cos  d  2  sin2  £/  =  y (b) 

Then  (a)  becomes  cos  z  —    cos  z9  —  y, (c) 

or  z  =  f(y\ 

This  expression  may  now  be  expanded  into  a  series  in  terms 
of  ascending  powers  of  y,  and  when  /  is  small  the  series  will 
converge  rapidly  if  #0  is  not  too  small. 

Maclaurin's  formula  applied  to  this  case  is  as  follows : 


24O  PRACTICAL  ASTRONOMY.  §  146. 

Differentiating  (c)  and  observing  that  when  y  =  o,  z  —  z» 
we  find  the  following  values  of  the  differential  coefficients  : 


ldz\  _       i 

\d]  ~~  sm      ; 


y 

Substituting  these  values  in  (d)  and  restoring  the  value  of 
y,  we  find 


cos  <   cos          .  cos  <    cos 

cot  *• 2 

(r*o^  tz?  c*os  (y \  ^ 
-sln~-o j  l(i+3  cot2^0)2  sin6 i/.    .     .     .     (240) 

In  this  equation  2  sin2^/,  2  sin4^/,  etc.,  are  expressed  in  terms 
of  the  radius.  The  equation  must  be  made  homogeneous  by 
introducing  the  divisor  sin  i"  where  necessary. 

cos  cp  cos  d 

1x84       - 


.      (24I) 

+     />     f+f^T        ty    \      I        •                                                   _     xi 
^     \^\JL       &Qj      O     j                             *                ft        "     • 

Then  we  have 

<p  =  3  ±  z  =F  -^^  ±  ^  =F  6^.     .     .     .     (242) 

146.  This  computation  is  made  very  simple  by  the  use  of 
table  VIII,  where  m  and  n  are  given  with  the  argument  /  ex- 
pressed in  time  (the  last  term,  Co,  is  seldom  used). 

As  A  and  B  will  be  constant  for  the  entire  series,  we  shall 
have, 

If     z^   zv  <sr3,  etc.,  z^,  are  the  observed  zenith  distances, 
mlt  m»  Ma,  etc.,  m^  the  corresponding  values  of  m  taken 
from  the  table, 


§  148.     LATITUDE   BY  CIRCUMMERIDIAN  ALTITUDES.         24! 

«„  «„  n,,  etc.,  n^  the  corresponding  values  of  «, 


<p  =  #  ±.  j^  =F  ^*»,  ± 

<?  =  <?  ±  *a  =F  -4«,  ± 

9?  =   tf  ±  2^  =F  ^*0M  ±  -# 

The  mean  of  these  equations  will  then  be 


*    -L        » 

=  <*  ±  - 


147.  It  will  be  observed  that  an  approximate  value  of  the 
latitude  is  required  for  computing  A.     When  the  observa- 
tions extend  on  both  sides  of  the  meridian  a  sufficiently  close 
approximation  may  always  be  obtained  by  taking  the  largest 
measured  altitude  and  calling  this  the  meridian  altitude  ;  or, 
better,  take  the  mean  of  this  in  connection  with  that  imme- 
diately preceding  and  following  it.     If  the  altitudes  are  all 
measured  on  one  side  of  the  meridian,  or  if  for  any  reason  a 
value  of  (p  has  been  used  which  proves  to  be  considerably 
in  error,  it  may  be  necessary  to  repeat  the  computation  of  A, 
using  for  cp  the  value  found  from  the  first  computation.     In 
that  case  only  the  correction  Am  need  be  computed  in  the 
first  approximation,  and  only  three  or  four  altitudes  reduced. 

148.  Let  us  now  examine  separately  the  terms  of  equation  (240)  in  order  to 
see  how  far  from  the  meridian  the  observations  may  be  extended  without  intro- 
ducing into  the  resulting  latitude  inadmissible  errors. 

Taking  the  last  term,  viz., 


/cos  cp  cos  <5\3o/  x2sinH/ 

I    .      .    -  jrl  K!  -f  3  C0t220)—  r    -ft-  =   Co, 

\sin  (q>  —  d)f  sin  i" 

for  any  given  values  of  (p  and  d,  we  can  compute  the  value  of  /,  for  which  this 


242 


PRACTICAL  ASTRONOMY. 


§I48. 


quantity  will  have  any  value,  as,  for  instance,  i".  We  readily  see  that  when 
the  zenith  distance  of  the  star  is  large  the  observations  may  be  extended  much 
further  from  the  meridian  than  when  it  is  small.  The  following  table  gives  the 
hour-angle,  for  which  this  term  has  the  value  i"  for  different  values  of  <p  and  d. 
Thus,  referring  to  the  table,  we  see  that  if  cp  =  40°  and  d  =  o,  then  t  =  40™  ; 
or,  in  this  case,  the  error  committed  in  neglecting  this  term  amounts  to  i"  only 
when  the  star  is  40™  from  the  meridian.  If  <p  =  40°  and  d  =  23°  about  the 
maximum  declination  of  the  sun,  then  /  =  2Om. 

LIMITING  HOUR-ANGLE  AT  WHICH  THE  THIRD   REDUCTION  AMOUNTS  TO  ONE 

SECOND. 


Declination  same  sign  as  Latitude. 

Declination  different  sign  from  Latitude. 

Latitude. 

80° 

70° 

60° 

50° 

40° 

30° 

20° 

10° 

0° 

10° 

20° 

30° 

40° 

50° 

60° 

70° 

80° 

0° 

135m 

90™ 

67m 

5Im 

40"*   29" 

20™ 

nm 

Dm 

nm 

20°» 

29m    40m 

51m 

67" 

9o"> 

I35m 

10 

128 

82 

59 

43 

32 

21 

II 

0 

i 

I 

20 

28 

37 

47 

59 

75 

Q6 

20 

118 

73 

35 

23 

12 

O 

ii 

2 

3 

28 

37 

46 

i6 

67 

82 

30 

107 

64 

42 

26 

14 

0 

12 

21 

29 

37 

46 

55 

64 

75 

40 

95 

54 

32 

16 

o 

14 

23 

32 

40 

47 

50 

04 

73 

50 

82 

42 

19 

0 

ib 

2b 

35 

43 

5 

I 

59 

67 

75 

60 

67 

27 

o 

19 

32 

42 

59 

6 

7 

75 

82 

70 

45 

0 

27 

42 

54 

04 

73 

82 

90 

96 

Let  us  now  consider  the  term 

/cos  cp  cos 
\sin  (q>  — 


•^1  cot  z0 


sin  i 


_, 

-  =  Bb. 


In  a  precisely  similar  manner  we  can  compute  the  limiting  values  of  /,  within 
which  this  term  is  less  than  i".  The  table  is  computed  in  this  way  ;  from  it 
we  find  that  in  the  first  of  the  above  cases  /  =  i6m  ;  in  the  second,  /  =  9™. 

LIMITING  HOUR-ANGLE  AT  WHICH  THE  SECOND  REDUCTION  AMOUNTS  TO  ONE 

SECOND. 


Declination  same  sign  as  Latitude. 

Declination  different  sign  from  Latitude. 

Latitude. 

80° 

70° 

60° 

50° 

40° 

30° 

20° 

10° 

0° 

10° 

20° 

30° 

40° 

50* 

60° 

70° 

80° 

0° 

67m 

39m 

27m 

2Im 

i6m 

1  2™ 

8™ 

5m 

Dm 

5m 

gm 

I2m 

i6m 

2Im 

2?m 

39m 

67m 

10 

54 

33 

24 

17 

13 

9 

5 

o 

8 

12 

*5 

19 

24 

32 

48 

20 

4« 

29 

20 

14 

10 

5 

0 

5 

12 

IS 

18 

23 

29 

40 

3° 

43 

26 

17 

II 

6 

0 

5 

9 

i 

2 

15 

18 

22 

28 

37 

4° 

38 

22 

13 

7 

0 

6 

10 

13 

16 

19 

23 

28 

36 

50 

33 

18 

9 

0 

7 

ii 

14 

J7 

21 

24 

29 

37 

60 

28 

12 

0 

9 

13 

i? 

20 

24 

2 

7 

32 

40 

70 

20 

0 

12 

18 

22 

26 

29 

33 

39 

48 

§  149-  LATITUDE  BY  CIRCUMMERIDIAN  ALTITUDES.  243 

* 

If  we  are  able  to  choose  our  own  times  for  observing,  we  can  always  make 
our  measurements  so  near  the  meridian  that  these  terms  may  be  neglected. 

As  i"  is  much  within  the  error  of  an  ordinary  sextant  measurement,  the 
limits  may  be  extended  somewhat  beyond  those  of  the  table  without  serious 
error.  We  may,  in  a  similar  manner,  determine  for  what  values  of  /  Co  or  Bn. 
will  have  the  values  o".i,  o".oi,  or  any  other  value. 

Lower  Culmination. 

149.  When  the  star  is  observed  near  the  meridian  at  lower 
culmination,  the  hour-angles  should  be  reckoned  from  the 
lower  branch  of  the  meridian.  This  is  equivalent  to  substi- 
tuting 180°  +  ^  in  the  formula  in  place  of  /.  We  then  have 

cos  z  =  sin  q>  sin  d  —  cos  g>  cos  d  cos  /. 

Writing,  as  before,  cos  t  =  i  —  2  sin9^/, 

,  » 

this  becomes 

cos  z  —  —  cos  (<p  +  tf)  -f-  cos  9  cos  #  2  sin2J/. 

Expanding  this  as  before,  and  remembering  that  for  lower 
culmination  we  have,  from  (235), 

*0  =  180°  -(<?  +  £), 
and  therefore     cos  2Q  =  —  cos  ((p  -f  6), 

we  readily  obtain 

,  cos  <p  cos  #2  sin2-!/  ,   /cos  (p  cos  tf 


or  z9  —  z  +  Am  +  Bn,     .....     (245) 

and  cp  =  180°  —  <$  —  (z  +  Am  +  Bn).      .     .    (246) 

This  formula  might  have  been  obtained  from  (240)  exactly 

as  (235)  is  from  (234),  viz.,  by  simply  changing  £  into  180°—  d. 

The  hour-angle  is  obtained  by  simply  taking  the  difference 


244  PRACTICAL   ASTRONOMY.  §  149. 

# 

between  the  chronometer  time  of  observation  and  of  culmi- 
nation.* 

Let  a  =  star's  right  ascension  =  sidereal  time  of  culmination ; 
JO  =  chronometer  correction,  +  when  chronometer  is  slow. 
Then     (a  —  A®)  =  chronometer  time  of  culmination. 

If  then  ®'  is  the  chronometer  time  of  any  observation, 

/=©'_(*_  J0) (247) 

Formula  for  Latitude  by  Circummeridian  Altitudes  of  a  Star. 


t  =  @    —  (a  - 
_  COS  q>  COS  $ 

A  — 

sin  z9 

2  sin4 

n  =  —. — 


9?  =  d  ±  (<sr  —  ^M  +  ^«),  upper  culmination  ; 

cp  =  180°  —  d  —  (z  -}-  Am-\-  Bn\  lower  culmination.  J 


Example  of  Latitude  by  Circummeridian  Altitudes. 

1873,  August  20.  a  AquilcB  observed  for  Latitude.  Observer  Boss. 

Instruments:  Sextant  and  Sidereal  Chronometer. 

Assumed  latitude  q>  =       49°  01' 
Assumed  longitude  L  =  -j-    ih  4im  i88 
Chronometer  correction  A®  =  —         22    50 
From  ephemeris,  right  ascension  of  star  a  =       igh  44™  37".  5 
Therefore  chronometer  time  of  culmination  =  a  —  A®  =       20     7    27  .5 


Star's  declination  8  =          8°  32'  n".5 


*  If  the  rate  of  the1  chronometer  is  appreciable  it  must  be  taken  into  account 
For  the  simplest  manner  of  doing  this  see  Art.  152. 


§  149-     LATITUDE  BY  CIRCUMMERIDIAN  ALTITUDES.         245 


<p  =  49°  01'. 
d  =    8    32  .2 

cos  <p  =  9.8168 
cos  8  =  9.9952 

log  A*  =  9.9992 
cot  z0  =     .0688 

*1ncr  R  —  O  ofiSo 

A  —    .9991 

locr  A  —  o.onnfi 

B  =  1.169 

The  observations  and  method  of  reduction  are  shown  in 
the  following  tabular  statement,  which  will  be  sufficiently 
explained  by  reference  to  formulae  (XIII). 


Sextant. 
zk. 

h. 

Chronometer. 
©'. 

t. 

T 

99°  5'  35" 

49°  32'  47".  5 

20h    ira  358 

-  5m  52s.  5 

2 

6   10 

33     5 

2    37 

4    50-5 

3 

7     5 

33  32  -5 

3    57 

3    30-5 

4 

7  55 

33   57  -5 

5      5 

2     22  .5 

5 

8   10 

34     5 

6    41 

-           4^-5 

6 

8     o 

34     o 

7    52 

+           24.5 

7 

7  50 

33   55 

8    5i 

I      23  .5 

8 

7  4° 

33   50 

9    47 

2    19  -5 

9 

7     5 

33  32  .5 

10    41 

3    13  -5 

10 

99   6  55 

49   33   27  .5 

20     12        0 

+4    32  -5 

m. 

Am, 

«.* 

Bn.*- 

h  -{-Am  —  Bn. 

V. 

vv. 

i 

67".8 

67".7 

".01 

".01 

49°  33'  55"-  2 

4.6 

21.  16 

2 

46  .0 

46  .0 

.01 

.01 

51  -o 

8.8 

77-44 

3 

24    .2 

24    .2 

56  .7 

3-i 

9.61 

4 

II    .1 

II    .1 

68  .6 

8.8 

77-44 

5 

I    .2 

I    .2 

66  .2 

6.4 

40.96 

6 

•  3 

•  3 

60  .3 

•5 

.25 

7 

3  .8 

3  -8 

58  .8 

I.O 

1.  00 

8 

10  .6 

10  .6 

60  .6 

.8 

.64 

9 

20    .4 

20   .4 

52  .9 

6.9 

47.61 

10 

40  .5 

40  .5 

49    33  68  .0 

8.2 

67.24 

Mean  h  =  49°  33'  59".  8 
Index  error  =•$•/=—  i  51  .5 
Eccentricity  =^£=  —  10  .1 
Refraction  r  =  —  47-3 


\yv\  =  343.35 

r  -  3".9 

r0  =  I  .3 


*  It  is  easy  to  see  in  advance  than  the  term  Bn  is  inappreciable  in  this  case. 
Jt  is  introduced  here  to  illustrate  the  method. 


246  PRACTICAL  ASTRONOMY.  §  150. 

Corrected  altitude  =  49°  31'  io".9 
Zenith  distance  z  =  40  28  49  .1 
Declination  d  =  8  32  n  .4 

Resuking  latitude  q>  =  49      i     o  .5  ±  i".3 

If  it  is  not  considered  necessary  to  reduce  each  observa- 
tion separately,  the  work  is  abridged  somewhat  by  the  fol- 
lowing process  [see  Art.  (146)] : 


Mean  of 
Index 
Eccentricity 
Corrected 

Refraction 
Corrected 
Zenith  distance 
Declination 
Latitude 

2/fc  =  99°    7'  14".  5 
I  =  -      3  43.  .0 

E  =   —              20   .2 

*h  =  99      3   ii  .3 
h  =  49    3i   35  -6 
Am'  —  -f-          22  .6 

47  -3 
h  =  49    31    10  .9 
z  =  40    28  49  .  1 
d  =    8    32   ii  .4 
<p  =  49      i     0.5 

Mean  of  m  =  22". 6  =  m' 
Am'  =  22".6 


150.  In  the  formulae  which  we  have  derived  for  circum- 
meridian  altitudes  we  have  supposed  the  declination  prac- 
tically constant  during  the  interval  of  observation. 

With  the  sun  this  is  not  the  case  ;  but  the  same  method  may 
be  used  if  we  take  for  d  the  mean  of  the  declinations  corre- 
sponding to  each  time  of  observation,  or,  what  is  practically 
the  same,  the  declination  corresponding  to  the  mean  of  the 
times.  It  is,  however,  better  to  reduce  each  altitude  sepa- 
rately for  the  purpose  of  estimating  the  accuracy  of  the  final 
result  and  as  a  partial  check  against  error  of  computation. 
If  formulae  (XIII)  are  used,  the  declination  must  be  inter- 
polated for  the  time  of  each  altitude ;  this  considerably  aug- 
ments the  labor  of  reduction.  This  additional  labor  may  be 
avoided  by  the  method  which  follows. 


§151-      CIRCUMMERIDIAN  ALTITUDES  OF   THE   SUN.          247 

Gauss    Method  of  Reducing   Cir  cummer  idian   Observations  of 

the  Sun. 

151.  In  this  method  the  hour-angle  is  reckoned  from  the 
point  where  the  sun  reaches  his  maximum  altitude  instead  of 
from  the  meridian.  The  meridian  declination  may  then  be 
used  in  reducing  all  of  the  observations. 

Let  #0  =  the  sun's  meridian  declination; 

d  =  the  declination  corresponding  to  hour-angle  t\ 
Ad  m  hourly  change  in  tf  given  in  the  Nautical  Al- 

manac, -\-  when  the  sun  is  moving  N.; 
t  —  the  hour-angle  given  in  seconds  of  time. 

Ad 
Then  -—  =  the  change  in  tf  in  one  second, 


Ad 
and  d  =  £0  +  t         .......    (248) 

Also,  since  8  =  f(t\ 


by  neglecting  terms  of  higher  order  than  the  first.    Then 

dd       cos  (p  cos  # 
9  =  *  +  $o  +  t-fi  ~      -^T—  -  •  2  sm  i*»  etc.     (250) 

The  peculiarity  of  the  process  is  in  the  method  by  which 

I  PL 

the  small  term  t  .  —j-  is  taken  into  account.     For  this  pur- 
pose we  determine  the  value  of  t  corresponding  to  the  maxi- 
mum value  of  h  by  placing   ,-  equal  to  zero  and  solving  for  t. 
Take  the  equation 

sin  h  =  sin  cp  sin  d  -f-  cos  (p  cos  d  cos  t. 


248  PRACTICAL  ASTRONOMY.  §  I  5  I. 

Differentiating  with  respect  to  k,  d,  and  t,  and  placing  ^-=  o, 
we  have 

.  dh  dd 

cos  h  -£-  —  (sm  cp  cos  3  —  cos  9?  sin  tf  cos  /)  -77 

-  cos  (p  cos  tf  sin  /  =  o.  (251) 

As  /  will  be  very  small,  no  appreciable  error  will  be  intro- 
duced by  making  cos  t  =  i,  when  the  above  equation  readily 
gives 

dd       cos  cp  cos  d 

-77  =  ^—         —^r  sm  t.      ...     (252) 
<afr        sin  ((p  —  d) 

In  this  t  is  the  hour-angle  of  the  sun  corresponding  to  the 
maximum  altitude.  To  distinguish  it  from  the  general  value 
of  t  call  it  /,  and  as  it  is  small  we  may  write 

d$        cos  cp  cos  8 

'-*--   ^n^r~-^  .....  (253) 

7  PL 

Substituting  this   value  of  -     in  equation  (250),  it  becomes 


s.n5       _ 


sin  ^0 


Since  ^  will  always  be  small  when  this  method  is  used,  let  us 
write 

sin  J/  =  %t,        whence         2  sin2  \  t  =  %t\ 
Then          2  sin9  \t  -  ty  =  %(f  -  2ty  +  /)  -  -J/ 


Passing  back  from  the  angles  to  the  sines  and  making  the 


§151.       CIRCUMMERIDIAN  ALTITUDES   OF   THE    SUN.          249 

terms  homogeneous  by  introducing  the  divisor  sin  \'  ',  equa- 
tion (254)  becomes 

cos  <p  cos  tf    2  sin2  \y 


cos  q>  cos  d    2  sin*  ^  (/  —  y) 
sin  £0  sin  i" 

cos  cp  cos  £    2  sins  ^  j/  . 

The  term  —  -  .  —  •  -  rr  is  always  very  small,  and  in 

sin  #0  sm  I 

the  solution  of  the  problem  as  given  by  Gauss  it  was  neg- 
lected. Its  computation  only  requires  one  additional  loga- 
rithm, and  is  therefore  very  simple  ;  but  in  reducing  sextant 
work  it  is  perhaps  an  excess  of  refinement  to  retain  it. 

We  now  require  a  convenient  formula  for  computing  y. 

Equation  (253)  may  be  written 


sn 

" 


Q 

y  15  sm  i"  —  —  —3.  -=-,     .     .     .     (256) 

cos  cp  cos  d    dt 

since  y  will  be  required  in  seconds  of  time. 

If  we  replace  dd  by  the  number  of  seconds  of  arc  which  d 
increases  in  one  hour,  and  dt  by  one  hour  expressed  in  seconds 
of  arc,  we  have 

dd          Ad 


dt  '     54000* 
Then  from  (256) 

sin  ^0  206265  sin 


(257) 


Ad 
y  =  [9.40594]  --  ................     (258) 


25O  PRACTICAL  ASTRONOMY.  §  152. 

It  will  frequently  be  accurate  enough  to  take  y  =  ^-j-. 

A 

y  is  added  algebraically  to  the  chronometer  time  of  cul- 
mination; the  result  is  the  chronometer  time  of  maximum 
altitude.  The  difference  between  this  and  the  chronometer 
time  of  observation  is  (t  —  y]. 

Formula  for  Latitude  by  Circummeridian  Altitudes  of  the  Sun. 

=    [940594]    ~A\ 

/  Z?  /I  T      \        .A. 

,    (XIV) 
2  sin"  Ut  — 


— 

sin  i"  sin  \ 


<p  =  z  +  #0  +  x*  —  Am  +  Bn. 


Correction  for  Rate  of  Chronometer. 

152.  If  the  times  are  recorded  by  a  chronometer  which  has 
a  large  rate,  the  hour-angle  used  in  formulae  (XIII)  and  (XIV) 
may  require  a  correction.  This  correction  can  be  applied  in 
a  very  simple  manner,  as  follows : 

Suppose  first  a  star  to  be  observed  by  a  sidereal  chro- 
nometer which  has  a  daily  rate  #©,  -f-  when  the  chronometer 
is  losing.  Then  24  actual  sidereal  hours  correspond  to  24h  — tf@, 
as  shown  by  the  chronometer,  and  all  hour-angles  given  in 
units  of  chronometer  time  will  be  in  error  in  a  like  ratio. 

Let  /  =  any  hour-angle  as  shown  by  the  chronometer; 
/'  =  true  value  of  the  hour-angle. 

*  x  may  always  be  neglected  without  serious  error  when  2o  is  not  too  small. 


152.       CORRECTION  FOR  RATE   OF  CHRONOMETER.  251 

24h  86400s 


t  ~~  24h  -  d®  ~~  86400s- 


n 


""  86400J 
Then  in  formula  (XIII)  we  shall  have  with  practical  accuracy 

If- 

sin  \t'  :  sin  i*  =  /':*; 
sin2  \t'  =  k  .  sin2  \t. 

The  factor  k  or  log  k  may  be  conveniently  tabulated  with 
the  argument  rate  ;  and  as  it  will  be  constant  in  any  series 
of  observations,  it  may  be  combined  with  the  factor^,  which 
will  then  be  computed  by  the  formula 

.  COS  cp  COS  d 

A    =    k  --  r-  -  .     '     .....        (260) 

sin  #0 

k  is  given  in  table  VIII,  C. 

If  a  star  is  observed  with  a  mean  time  chronometer  whose 
rate  is  <5T,  the  factor  Vk  will  convert  the  chronometer  inter- 
vals into  mean  time  intervals  ;  we  then  require  the  factor 
[A*  —  1.00273791  to  convert  these  mean  time  intervals  into 
sidereal  intervals.  The  formula  for  computing  A  will  then  be 

,  cos  cp  cos  8 

A  —  k}£  -  f-  -  ,  .....     (261) 
sin  z9 

where  log  //  —  .0011874. 

If  the  sun  is  observed  with  a  mean  time  chronometer  the 
intervals  of  the  chronometer  corrected  for  rate  will  not 
correspond  exactly  to  the  solar  intervals,  as  these  will  be 
apparent  time  intervals. 

*  See  Art.  93. 


252  PRACTICAL  ASTRONOMY.  §  152. 

If  we  let  tiE  =  the  increase  of  the  equation  of  time  in 
one  day,  then  (one  apparent  solar  day)  =  (one  mean  solar 
day)  —  SE,  and  $T  -  -  $E  =  the  chronometer  rate  on  ap- 
parent time,  k  will  then  be  given  by  the  formula 

• 


k>  = 


6  T- 

86400 


Finally,  if  the  sun  is  observed  with  a  sidereal  chronometer, 
we  must  introduce  the  factor  —  to  convert  the  sidereal  inter- 
vals into  mean  time  intervals. 

The  log  -  —  9.9988126. 

The  formulas  for  'the  four  cases  are  then  as  follows  : 


k  = 


r 


(XV) 


86400J  86400 

_                                                                          .  cos  q>  cos  6" 
Star  with  sidereal  chronometer,      A  =  k r- ; 

sin  ZQ 

n  _  cos  <p  cos  d 
Star  with  mean  time  chronometer,  A  =  [0.002375]^ r- ; 

sin  ZQ 

,,  cos  <z>  cos  8 

Sun  with  mean  time  chronometer,  A  =  k  —         ; 

sin  z0 

Sun  with  sidereal  chronometer,       A  =  [9.997625]^' --—  '• . 

sin.  #o 

k  and  k'  are  taken  from  table  VIII,  C. 

Example.     Determination  of  latitude  by  circummeridian  altitudes  of  the  sun. 
1869,  July  24th.  Des  Moines,  Iowa.  Observer  Harkness. 

Instruments:     Sextant  and  Mean  Time  Chronometer. 
The  declination,  equation  of  time,  etc.,  are  taken  from  the  ephemeris  for  the 


§152.  CIRCUMMERIDIAN  ALTITUDES  OF  SUN.  253 

instant  of  the  sun's  meridian  passage  at  Des  Moines  =  ih  6m  16*  apparent 
time  at  Washington. 

Assumed  Latitude     _          <p  =       41°  35'.$ 

Longitude  L  =  +    ih    6m  i6« 

Chronometer  correction  4  7^  =  —    6    18       8.9 
From  ephemeris,  S  =        19°  46'    i6".i 

4$  =  —  31  .94 

Equation  of  time  E  =  -j~  °m  I28.o 

Semidiameter  S  =  15'   47".  2 

Equatorial  hor.  parallax      TT  =  8  .44 

Computation  of  A  and  B. 

<?  =  4i°35-5  cos  =  9.8738 

S  =  19  46.3  cos  =  9.9736  log  A1  =  0.5544 

zQ  =  21   49.2  cosec  —    .4298  cot  z0  =    .3975 

,4  =  1.893  log  ,4=    .2772  log  B  =    .9519 


B  =  8.95 

Computation  of  y.  Computation  of  x*.                INDEX  ERROR. 

On  arc.  Off  arc. 

Constant  log  =  9  4059  2  sin'2  \y  _  „                  29'      5"  33'     60" 

log  A8  =  i.5043n  sin  i"                                     10                 40 

i  TO                 50 

log—   =9.7228  x  =    .02 J 

29'      8".  3  33'     50" 


log  y  =    .6330,1  /  =  -f-  2'  20".  i 

y  =  —  48-3 

For  the  chronometer  time  of  culmination  we  have 

Equation  of  time  E  =       oh    6m  I28.o 

AT  =  -  6    18       8.9 

y=-  4-3 


Chronometer  time  of  max.  alt.  =       6h  24m  i6*.6 

The  difference  between  this  quantity  and  the  observed  time  T  is  the  quantity 
(t-y). 

*  In  reducing  sextant  observations  x  may  always  be  disregarded. 


254  PRACTICAL   ASTRONOMY. 

The  observations  and  reductions  are  now  as  follows : 


§   152- 


Upper  limb- 


Lower  limb- 


( 

.. 

Sextant 

2/1. 

h. 

Chro- 
nometer 
T. 

't-y. 

m. 

Am. 

w. 

Bn. 

>%  +  ^w 

-  Bn 

[l 

i360i7'45" 

68°  8'52".5 

6h  7m5i8-5 

i6m25".i 

529".! 

iooi'x.6 

.68 

6".  i 

68»25'28".o 

r 

20  10 

10    5 

8   32 

*5   44  -6 

486   .5 

920   .9 

•58 

5   -2 

20     .7 

(3 

22  2O 

II  IO 

9     9-5 

15      7  .IJ448    .6 

849     .2 

.48 

4    -3 

14   .9 

ii 

135  23  10 
25  10 

29  30 

67  4i  35 
4235 
44  45 

10     II   .3 

1°  55  -3 
6  12      7  .0 

14      5  -3 
13    21  .3 
12     9  .6 

389    -6 
350   .1 
290   .3 

737    -5 
662    .7 

549    -5 

•37 
•30 
.20 

3    -3 

:* 

67  53  49   -2 
35   -o 
52   -7 

Mean  h  Q  =  68°  25'  2i".2 

Semidiameter  S  =  —  15  47  .2 
Refraction  r  =  —  21  .6 
Parallax  /  =  -f-  3  -i 

Index  cor.  \I  =  -f-  i  10  .4 
Eccentricity  \E  —  -j-  14  .8 


Corrected 
Mean 


®  -  67°  53'  45"-6 

+        !5  47  -2 

21  .8 

+  3-2 

-f-          i   10  .4 
+  14  .8- 

68°  10'  39".4 


A  =  68°  10'  40".  7 
h  =  68°  10'  40". o 
z0  =  21   49  20 
5  =  19  46  16 
Resulting  latitude  cp  =  41°  35'  36" 


The  observations  of  the  above  series,  it  will  be  noticed, 
were  all  taken  before  the  sun  reached  the  meridian,  and  so 
far  from  the  meridian  that  the  term  Bn  has  a  very  appreci- 
able value.  It  is  a  little  better  to  take  the  observations  near 
the  meridian  when  practicable,  as  then  small  errors  in  AT 
will  produce  less  effect  on  the  resulting  latitude.  (See  Art. 
144.) 

The  above  observations  may  be  reduced  by  the  method 
of  Art.  146  if  it  is  not  considered  necessary  to  compare  the 
individual  results.  The  labor  is  considerably  less,  as  will  be 
seen  by  the  following : 

Mean  of  chronometer  times  •         =         6h    9™  47*.  8 
AT  =  -    6    18       8.9 

True  mean  time  —        23    51     38 .9 

Longitude  from  Washington     L  =         i     6     16 
Washington  mean  time  =         o   57     54-9 


§  I$2.  CIRCUMMERIDIAN  ALTITUDES   OF  SUN.  2  $5 

The  declination  of  the  sun  is  now  to  be  taken  from  the  ephemeris  for  this 
mean  time  of  observation,  instead  of  the  instant  of  meridian  passage  as  in  the 
previous  method. 

Thus  S  =  19°  46'  23".  8; 

E  =          6rai2s.o. 

This  value  of  E  is  now  the  mean  time  of  the  sun's  meridian  passage.  For 
the  chronometer  time  we  have 

E  =       oh     6m  I2«.o; 

JT=  —  6    18       8.9; 

'Chronometer  time  of  the  sun's  meridian  passage  —       6    24     20 .9. 


Chronometer 

T. 

t. 

m 

n. 

6h 

7m 

51' 

'•5 

i6m 

2$ 

'•4 

533' 

'•7 

.69 

8 

32 

.0 

15 

48 

•9 

491 

.0 

•59 

9 

9 

•  5 

15 

II 

•4 

452 

•9 

.49 

10 

ii 

•  3 

14 

9 

.6 

393 

.6 

.38 

10 

55 

•3 

13 

25 

.6 

353 

•9 

•31 

6 

12 

7 

.0 

12 

13 

•9 

293 

•7 

.21 

Am   =794  .7         Bn   =  3".9 

The  number  of  observations  on  the  two  limbs  being  the  same,  the  serm- 
diameter  will  be  eliminated  by  taking  the  mean  of  the  individual  values. 

Mean  of  sextant  readings  =  ih  =  135°  53'  oo".8 
Index  correction  =  /  =  -f-  2  20  .8 
Eccentricity  E  =  +  29  .7 


Corrected  reading  =  135°  55'  51". 3 

k  =    67   57  55  .6 

Refraction  r  =  —    •        21  .7 

Parallax  /  =  +  3-2 

-f  Am  =  +     13   14  .7 

-  Bn   =  -  3  .9 


Corrected  altitude  =  68°  10'  47".9 

z0  =  21   49   12 

d  =  19   46   24 

•Resulting  latitude        q>  =  41°  35'  36" 


2 $6  PRACTICAL   ASTRONOMY.  §  153. 

The  rate  of  the  chronometer  was  8T  —  —    '.47 

The  daily  increase  of  the  equation  of  time  dE  =  -j-     .63 

5  r  -  SE  =  -  i  .  10 

Therefore  the  log  k  =  9.999989.     (See  Art.  152.) 

The  correction  for  rate  is  therefore  absolutely  inappreciable. 


Fifth  Method. 

J53-  By  Polaris  observed  at  any  hour-angle.  We  have  al- 
ready seen  (method  third)  how  the  latitude  may  be  obtained 
by  an  altitude  of  a  star,  observed  in  any  position.  We  have 
also  applied  the  formulae  deduced  to  a  series  of  altitudes  of 
Polaris. 

A  more  convenient  formula  than  the  one  there  used  is  ob- 
tained by  expanding  the  expression  for  the  latitude  into  a 
series  in  terms  of  ascending  powers  of  the  polar  distance.  The 
latter,  in  case  of  Polaris,  being  at  present  only  about  i°  20', 
the  series  will  converge  rapidly,  and  a  very  few  terms  give 
an  approximation  sufficiently  accurate  for  every  practical 
purpose. 

Let  /  =  90°    -  d  =  the  polar  distance; 

q>  =      h  —  x. 

Then  x  is  the  correction  which  is  to  be  applied  to  the  meas- 
ured altitude — corrected  for  refraction — to  produce  the  lati- 
tude,    x  can  never  be  greater  than  /. 
Substituting  these  values  in 

sin  h  =  sin  cp  sin  d  -|-  cos  cp  cos  8  cos  tt 
it  becomes 

sin  h  —  sin  (h  —  x)  cos/  +  cos  (h  —  x)  sin/  cos  t.     (a) 
Expanding  sin  (h  —  x)  and  cos  (h  —  x)  by  Taylor's,  and 


§153-  LATITUDE  BY  POLARIS.  257 

sin  /  and  cos  /  by  Maclaurin's  formula,  we  have,  as  far  as 
terms  of  the  order/4  and  x\ 

sin  (Ji—x)  =  sin  h  —  x  cos  //—  ^x*  sin  h  +  \x*  cos  h  -\-  -fax*  sin  h\ 
cos  (h—  ;r)=cos  h-\-  x  sin  h—  \x*  cos  h  —  \x*  sin 

sin  p=p-W*\ 

cos/  =  i  -  i/2  + 


Substituting  these  values  in  (a),  we  readily  obtain 

x  —  p  cos  /  —  i(V  —  2xp  cos  /  -|-  /')  tan  Jl 

+  K-*"3  —  3*>  cos  ^  +  3-r/  —  /3  cos  /)  ( 

—  4*3/  cos  ^+6^2/2—  4^>3  cos  ^+/4)  tan  >^. 


Which  contains  all  terms  in/  and  ^r,  from  the  first  to  the 
fourth  orders  inclusive,  x  must  now  be  determined  from 
(b)  by  successive  approximations.  For  the  first  approxima- 
tion let 

x  =  p  cos  t  .........     (c) 

Substituting  this  value  in  the  second  term  of  (b)  and  retain- 
ing terms  of  the  order/2,  we  find  for  the  second  approxima- 
tion 

x  —  p  cos  t  —  i/8  sin2  t  tan  h  .....     (d) 

Substituting  this  value  in  the  second  and  third  terms  of  (&) 
and  retaining  terms  of  the  order/3,  we  find  the  third  ap- 
proximation, viz., 

x  =  /  cos  /  —  -J/2  sin2  /  tan  h  -f-  -J/3  cos  /  sin3  1.    .     (e) 
Similarly  for  the  fourth  and  final  approximation, 

x  =  p  cos  t  —  ^/2  sin2  /  tan  h  +  i/3  cos  t  sin2  / 

—  i/4sin4  /  tanYz  +  ^/4(4—  Qsin2  /)  sin2  /tan  /*.( 


PRACTICAL   ASTRONOMY.  §  153. 

As  x  and  /  will  be  expressed  in  seconds  of  arc,  the  series 
must  be  made  homogeneous  by  multiplying/2  by  sin  i",/3  by 
sin2  i",  and/  by  sin3  \"  . 

Then  the  expression  for  the  latitude  is 

cp  —  h—p  cos  t  +  i  /2  sin  i  "  sin2  /  tan  7z 

-  \p*  sin2  i"  cos  /  sin2  1  +  -J/4  sin3!77  sin4  /  tan3  h 

-  ^/  sin3  \"  (4  —  9  sin2  /)  sin2  /  tan  h.  (263) 


Let  us  now  examine  separately  the   last  three  terms  of 
(263)  in  order  to  see  when  they  may  be  neglected. 
Let  us  write  the  last  term  equal  to  u,  viz., 

u  =  ^V/4  sin8  l"  (4  ~~  9  sm2  *)  sm2  ^  tan  ^* 


Forming  the  differential  coefficient  of  u  with  respect  to  /, 
placing  it  equal  to  zero  in  order  to  determine  what  value  of 
/  will  make  u  a  maximum,  we  find 

sin  t  cos  /  (2  —  9  sin2  /)  =  o; 
from  which 

sin  /  =  o  ;         cos  t  =  o  ;        sin2  /  =  f  . 

The  last  of  these  corresponds  to  a  maximum,  as  will  be 
found  by  substituting  this  value  in  the  second  differential 
coefficient. 

The  maximum  value  of  this  term  is  then  found  to   be 
(/  being  i°  20') 

u!  =  c/'.ooii  tan  h. 

It  will  therefore  always  be  inappreciable. 

The  next  term,  viz.,  \p"  sin3  \"  sin4  /  tan3  h,  is  a  maximum 
when  sin  /  =  i. 

Its  greatest  value  is  therefore  o//.oo76  tan3  h, 


§153-  LATITUDE   BY  POLARIS.  259 

This  term  will  then  be  only  o".oi  in  latitude  48°,  and  o".i 
in  latitude  67°.  It  may  therefore  always  be  neglected  when 
the  instrument  used  is  the  sextant. 

Writing  v  —  \p*  sin2  i"  cos  /  sin2  /, 

dv 
forming  ^-,  placing  it  equal  to  zero,  we  readily  find  that  v 

is  a  maximum  when  sin2  t  =  f.  The  maximum  value  of  this 
term  will  then  be  o".333.  If  then  we  drop  this  term  with 
those  which  follow,  the  error  introduced  in  this  way  will 
seldom  amount  to  half  a  second,  and  will  generally  be  much 
smaller  as  the  maxima  values  of  the  different  terms  occur  for 
different  values  of  t. 

Therefore  for  determining  the  latitude  by  Polaris  by  sex- 
tant observation, 


t  =  &  -  (a  -  J&);  1 

cp  —  h  —  p  cos  t  +  [4.384S4]/2  sin2  t  tan  k.  ) 

Let  us  apply  this  method  to  the  example  solved  in  Art.  143. 
We  have  given  — 

From  Nautical  Almanac.  By  Observation. 

a=    ihi5m6s.o  h=    39°  33'  38".  8 

£  =  88°  41'  6".  2  ©'  =    ioh45m   7".4 

Therefore/  =        4733".  8  A®  —  -f-  1  .5 

Therefore  /  =  142°  30'  43".  5 

constant  log  4.38454 

log/  =  3-675210  log/2  7.35042 

cos/  =  9.899537n  sin2/  9.56866 

tan  h  9.91704 

First  correction      —  i°  2'  36".  2        log  =  3-574747n 

Second  correction  -f  16  .6  log  2d  cor.  =  1.22066 

Therefore  cp  =  40°  36'  31".  6 


260  PRACTICAL  ASTRONOMY.  §  154. 

We  find  the  third  correction  to  be  o".24,  which    makes   the 
•value  of  <p  agree  exactly  with  the  value  before  found  (Art. 

143)- 

Tables  have  been  prepared  with  the  design  of  abridging 
this  computation,  but  the  direct  application  of  the  formula 
is  so  simple  that  tables  are  of  no  great  advantage,  especially 
if  the  third  and  fourth  corrections  are  not  required. 

Correction  for  Second  Differences. 

154.  When  a  series  of,  say,  ten  altitudes  is  observed,  if  the  measurements 
are  made  in  quick  succession,  so  that  the  arc  of  the  circle  in  which  the  apparent 
motion  of  the  star  takes  place  does  not  differ  appreciably  from  a  straight  line, 
then  the  mean  of  the  observed  altitudes  will  be  the  altitude  corresponding  to 
the  mean  of  the  times.  If,  however,  the  deviation  from  a  straight  line  is  ap- 
preciable, this  mean  altitude  will  require  a  correction  which  may  be  obtained  as 
follows: 

Let  /i,    ti,   /3,  .  .  .    /n  be  the  times  of  observation; 

hi,  hi,  /z3,  .  .  .  h^  be  the  observed  altitudes; 


hQ  =  the  altitude  corresponding  to  the  time  A>; 
Atl  =  A,  —  ti,  from  which  /<,  =  /i  +  ^^ 
At*  —  t0  —  /2,  t0  =  /a  +  4t0  ,  ,„ 

At^  =  A,  -  /n,  /0  =  ta  +  Atv 

Then  h0  =  /(A,)  ;         hi  =  /(6)  ;         /&n  =  /('n)  ; 

from  0),  hi  =  /(/«  -  Ati\  .  .  .  hn  =  /(/0  -  At^)  .......     (c) 

Expanding  these  expressions  by  Taylor's  formula,  we  find 


§154-  CORRECTION  FOR  SECOND  DIFFERENCES.  26 1 

The  mean  of  these  values  will  be 

Ai  -f-  A3  +  •  •  •  +  An   ,   dho  At\  +  At?  +  . . .  +  Atn 

i  a2Aj  2^r 


2  dt< 


From  the  values  At^  At*,  etc.,  by  (^),  the  term  multiplied  by  -— -  will  be  zero; 

at 

but  as  the  quantities   At\  ,*  Zf^2  ,  etc.,  will  all  be  plus,  the  term  multiplied  by 

d*h 

—f  will  not  be  zero.     It  should  always  be  taken  into  account  when  large  enough 

to  be  appreciable. 

To  determine  — ^  we  differentiate  the  equation 


sin  h  =  sin  q>  sin  d  -j-  cos  <p  cos  d  cos  /; 
when  we  readily  find 


cos  q>  cos  /cos  G)  cos  6"  \2  . 

s/0  +  sin'/otanA.     . 


(265) 


And  since  cos  A  =  sin  z,  this  equation  becomes 

-^  =  —  A  cos  A,  +  A*  sin2  /„  tan  A0 (266) 


The  quantities  Jh,  Ati,  etc.,  will  be  expressed  in  seconds  of  time  ;  they  must 
be  reduced  to  arc  by  multiplying  by  15.  Also,  i$At*,  etc.,  must  be  multiplied  by 
sin  i"  in  order  to  make  formula  (264)  homogeneous.  The  last  term  will  there- 
fore be  multiplied  by  i(i5)2  sin  i",  the  logarithm  of  which  is  6.73673  —  10. 
Therefore  formula  (264)  becomes 


(267) 


262  PRACTICAL  ASTRONOMY.  §  1 55 

As  an  example,  we  may  apply  formula  (267)  to  the  observations  of  Polaris 
given  in  Art.  143,  where  we  have 


At,  = 

I23M 

4/i2  =  15227.6 

4/,= 

71    .4 

At?  —    5098.0 

4/3  = 

5-4 

At?  =        29.2 

4/4    = 

—    42  .6 

47?  =    1814.8 

4/5    = 

-  157  -6 

At?  =  24837.8 

Mean  =    9401.5  log  =  3.9732 

Ji  i. 
By  formula  (265),  with  the  data  given  in  Art.  143,      log  -^  =  8.2898 

constant  logarithm  =  6.7367 
Correction  =  —  o".io  log  =  8.9997 


We  may  in  a  manner  precisely  similar  derive  the  correction  to  be  applied  to 
the  mean  of  the  times,  to  obtain  the  time  corresponding  to  the  mean  of  the 
zenith  distances:  this  may  be  more  convenient  in  certain  cases. 

The  necessity  for  applying  a  correction  for  second  differences  may  generally 
be  avoided  by  dividing  a  long  series  of  observations  into  two  or  more  parts, 
neither  of  which  shall  embrace  an  interval  of  time  long  enough  to  require  such 
correction.  This  proceeding  has  the  advantage  that  in  reducing  the  two  halves 
of  the  series  separately  they  will  mutually  check  each  other. 

155.  The  methods  of  determining-  time  and  latitude  which 
have  been  given  in  this  chapter  are  especially  adapted  to 
the  requirements  of  the  explorer.  The  observations  can 
generally  be  obtained  more  conveniently  at  night,  and  both 
time  and  latitude  will  be  required.  From  the  observed  time 
the  longitude  will  be  obtained,  as  will  be  explained  more 
fully  hereafter.  As  we  have  already  shown,  the  time  will  be 
best  determined  by  observing  two  stars,  one  east  and  one 
west  of  the  meridian,  both  as  near  the  prime  vertical  as  prac- 
ticable. 

The  latitude  will  generally  be  most  conveniently  deter- 
mined in  the  northern  hemisphere  by  observing .  Polaris 


§155-  GENERAL   REMARKS.  263 

north,  and  another  star  south,  by  circummeridian  altitudes. 
Then,  with  the  best  attainable  approximation  to  the  latitude, 
the  time  can  be  computed  by  the  method  of  Art.  125.  With 
this  value  of  the  time  the  correct  value  of  the  latitude  may 
then  be  determined  by  (XIII)  and  (XVI),  and  if  this  differs 
much  from  the  assumed  latitude  the  time  must  be  recom- 
puted. In  extreme  cases  it  may  be  necessary  to  recompute 
the  latitude,  but  with  proper  care  this  need  not  often  occur. 

As  a  survey  of  the  line  of  travel  is  generally  made  by 
means  of  a  compass  and  odometer  (which  is  a  little  instru- 
ment for  recording-  the  number  of  revolutions  of  a  cart- 
wheel), the  observer  always  knows  his  position  approxi- 
mately. The  same  process,  essentially,  is  followed  at  sea, 
where  the  approximate  place  of  the  vessel  is  always  known 
from  the  "  dead  reckoning,"  which  is  the  course  as  indicated 
by  the  compass  and  log. 

The  methods  of  this  chapter  are  those  which  are  most  con- 
venient and  useful  in  practice.  On  land,  where  the  observer 
has  a  certain  degree  of  choice  as  to  time  of  observation  and 
methods,  and  where  the  results  must  have  a  considerable 
degree  of  accuracy  to  be  of  any  value,  it  will  seldom  be  de- 
sirable to  employ  others.  At  sea,  however,  the  case  is  some- 
what different.  It  sometimes  happens  that  the  determina- 
tion of  the  place  of  the  vessel  is  of  the  greatest  importance 
when,  from  cloudy  weather  or  other  causes,  observations 
cannot  be  obtained  which  are  suitable  for  the  employment 
of  the  methods  of  this  chapter.  Further,  a  high  degree  of 
accuracy  is  not  required  for  purposes  of  navigation.  Vari- 
ous methods  of  determining  the  place  of  a  vessel  are  there- 
fore given  in  works  on  navigation,  in  order  that  the  mariner 
may  be  in  a  position  to  utilize  any  data  which  he  mav  obtain. 

It  can  readily  be  seen  that  by  varying  the  conditions  a 
great  variety  of  solutions  of  the  problem  may  be  obtained. 
Some  of  these  are  exceedingly  elegant  from  a  mathematical 


PRACTICAL  ASTRONOMY.  §  155. 

point  of  view.  Such,  for  instance,  is  the  method  given  by 
Gauss  for  determining  both  the  time  and  latitude  from  obser- 
vation of  three  stars  at  the  same  altitude.  Thus  if  //  is  the 
common  altitude,  tf,  £',  d"  the  declinations,  t,  t  -\-  A,  t  -{-  A' 
the  hour-angles  of  the  three  stars  respectively,  we  have 

0 

sin  h  =  sin  cp  sin  tf     -f-  cos  cp  cos  #    cos  / ;  ) 

sin  h  =  sin  cp  sin  $'  -\-  cos  cp  cos  tf'  cos  (/  +  A );  v  (268) 

sin  ^  =  sin  ^  sin  tf"  +  cos  cp  cos  tf"  cos  (t  -j-  A7).  ) 

Three  equations  from  which  t  and  cp  may  be  found.  Further 
than  this,  as  there  are  three  equations,  we  can  also  determine 
h  from  them,  so  that  the  altitude  need  not  be  measured  at 
all,  but  only  the  instant  of  time  observed  when  each  star 
reaches  the  altitude  //.  If,  however,  the  altitude  is  measured 
by  the  instrument,  this  process  shows  the  error  of  the  instru- 
ment, thus  giving  us  one  equation  for  determining  the  eccen- 
tricity by  Art.  116. 

If  three  altitudes  of  the  same  star  are  measured,  a  similar 
process  gives  us  three  equations  for  determining  the  latitude, 
hour-angle,  and  declination  of  the  star. 

Also,  it  is  evident  that  two  measured  altitudes  either  of  the 
same  star  or  of  different  stars  will  give  two  equations  of  the 
form  of  (268),  from  which  the  latitude  and  hour-angle  may 
be  determined.* 

A  variety  of  cases  may  also  be  considered  in  which  the 
measured  quantity  is  the  azimuth  of  a  star,  or  three  different 
altitudes  of  the  same  star  and  the  differences  of  the  azimuths, 
or  the  data  may  be  varied  in  many  ways  ;  but  these  solu- 
tions are  of  little  practical  value. 


*  For  a  solution  of  this  problem   graphically,   see  Captain  Sumner's  New 
Method  of  Determining  the  Place  of  a  Ship  at  Sea. 


156-    PROBABLE   ERROR   OF  SEXTANT   OBSERVATION.      265 


Probable  Error  of  Sextant  Observations. 

156.  In  all  instrumental  measurements  the  error  of  the  result  obtained  con- 
sists of  two  parts:  first,  that  due  to  the  observer;  and  second,  that  due  to  instru- 
mental and  other  sources  with  which  the  observer  has  nothing  to  do.  When  the 
instrument  employed  is  the  sextant,  the  latter  consists  for  the  most  part  of  the 
various  undetermined  errors  noticed  in  Articles  114-117.  In  any  given  series 
of  observations  these  affect  all  alike,  and  therefore  nothing  is  gained  in  this 
direction  by  increasing  the  number  of  individual  measurements. 

With  the  first  class,  however,  the  case  is  different.  These  form  the  accidental 
errors  of  observation,  and,  as  they  occur  in  accordance  with  the  law  of  least 
squares,  their  effect  diminishes  with  an  increase  in  the  number  of  measure- 
ments. 

Let  A'o  =  the  probable  error  of  the  mean  of  a  series  of  observed  altitudes; 
A>i  =  the  error  due  to  the  observer,  not  including  personal  equation; 
A"  2  =  the  error  due  to  instrument  and  causes  other  than  the  observer. 


Then,  by  Art.  16,  A>ft  =   VAY+  AY2  .........     (269) 

Thus  if  the  observer  could  do  his  part  perfectly,  he  could  never  diminish  the 
probable  error  of  a  single  series  below  AV 

The  values  of  .RQ,  AY  and  R*  for  a  given  instrument  and  observer  may  be 
determined  by  methods  which  we  have  already  employed. 

Thus  (Art.  132)  we  have  found  for  the  probable  error  of  the  time  determined 
by  a  series  of  ten  double  altitudes  of  the  sun,  AY  —  ±  M4.  The  corresponding 
error  in  the  double  altitude  zh  is  found  by  the  differential  formula,  viz., 

d2h   . 

A-2.fi  =  —j-4t, 
dt 

and  for  this  case  we  have  found   -  =  .640. 

dzk 

Therefore  A<2.h  =  8 


.649 
From  the  latitude  observations  (Art.  149)  we  have  found  2"  .  6  =  RI"  . 

By  a  discussion  of  the  ninety  individual  measurements  of  altitude  employee 
in  the  investigation  of  the  eccentricity  of  the  sextant  (example,  Art.  116),  Prof. 
Boss  finds  the  probable  error  of  a  single  measurement  of  double  altitude  to  be 
±  14",  and  of  the  mean  of  ten  measurements  ±  4".  4  ==  AY  From  the  solu- 
tion of  the  equations  of  condition  of  the  same  example  we  found  for  the  probable 


266  PRACTICAL  ASTRONOMY.  §  156. 

error  of  a  single  equation  RQ  =  5". 9.  Therefore  by  equation  (269)  R*  —  3". 93. 
Thus  the  instrumental  probable  error  is  nearly  equal  to  the  observer's  probable 
error  of  a  mean  of  ten  measurements. 

If  now  we  assume  the  probable  error  of  a  single  measurement  to  be  ±  14" 
as  above,  we  have  for  the  observer's  probable  error  of  the  mean  of  m  measure- 
ments, by  equation  (25), 


and  the  total  probable  error  R*  —  <y  — \-  15  .45. 

If     m  =  i,     Ro  =  14".  5;         m  =  10,     RQ  =  5.9;         m  =    50,     R*  —  4.4; 
m  —  5,     RV  =    7   .4;         m  =  20,     A'0  =  5.0;         m  =  100,     R$  =  4.2. 

Thus  it  appears  that  with  a  skilled  observer  almost  nothing  is  gained  by  ex 
tending  the  number  of  observations  of  a  given  series  beyond  ten.  Instead, 
therefore,  of  multiplying  observations  in  the  same  circumstances,  when  accuracy 
is  desired,  the  circumstances  must  be  varied  with  a  view  to  eliminating  the  in- 
strumental errors. 

Thus  for  good  results  a  determination  of  time  or  latitude  should  never  depend 
tn  a  single  series,  no  matter  how  carefully  made  or  how  elaborately  the  instru- 
mental errors  have  been  investigated.  Latitude  should  be  determined  by  both 
Jiorth  and  south  observations,  giving  both  equal  weight,  no  matter  whether 
determined  from  an  equal  number  of  measurements  or  not.  In  like  manner 
time  should  be  determined  from  observations  both  east  and  west  combined  with 
equal  weights.  (See  also  Harkness,  Washington  Observations,  1869,  Appendix  I, 
page  *i.) 


CHAPTER  VI. 

THE  TRANSIT  INSTRUMENT. 

157.  When  the  time  is  required  with  extreme  accuracy, 
as  in  a  careful  determination  of  longitude,  the  methods  of 
the  preceding  chapter  are  not  adapted  to  the  purpose.  The 
instrument  used  will  then  be  the  transit. 

The  common  form  of  transit  instrument  consists  essentially 
of  a  telescope  attached  to  an  axis  perpendicularly.  As  it 
revolves  with  the  axis  the  line  of  collimation  produced  to 
the  celestial  sphere  describes  a  great  circle.  The  instrument 
is  generally  mounted  so  that  this  great  circle  is  the  meridian, 
and  it  is  used  in  connection  with  the  sidereal  clock  or  chro- 
nometer for  determining^the  instant  of  a  star's  transit  over  the 
meridian.  If  our  clock  is  accurately  regulated  to  show  side- 
real time,  such  an  observed  transit  gives  us  at  once  the  star's 
right  ascension,  the  latter  being,  as  we  have  seen,  the  same 
as  the  sidereal  time  of  culmination.  If,  however,  we  observe 
a  star  whose  right  ascension  is  already  known,  this  process 
gives  us  the  error  of  the  clock.  The  field-transit  mounted 
in  the  meridian,  with  which  we  are  at  present  more  par- 
ticularly concerned,  is  always  used  for  this  latter  purpose. 

Theoretically  the  instrument  may  be  used  in  any  vertical 
plane.  It  is  sometimes  used  in  the  plane  of  the  prime  ver- 
tical for  finding  the  latitude,  or  in  a  fixed  observatory  for 
finding  the  declinations  of  stars.  When  speaking  of  the 
transit  instrument  simply  we  understand  it  to  be  mounted 
in  the  meridian. 


208 


PR  A  C  TIC  A  L   AS  TRONOM  Y. 


§158. 


FIG.  «6. 


$158.  THE    TRANSIT  INSTRUMENT.  269 


Description  of  the  Instrument. 

158.  The  ransit  instrument  designed  for  a  fixed  observa- 
tory, where  it  is  permanently  mounted,  is  much  larger  and 
more  complete  than  one  designed  for  use  in  the  field,  where 
it  must  be  transported  from  place  to  place.  The  transit- 
circle  of  the  Washington  observatory,  for  instance,  has  a 
telescope  of  twelve  feet  focal  length,  the  aperture  being  eight 
and  one  half  inches  ;  it  is  mounted  on  massive  piers  of  marble, 
which  rest  on  a  foundation  of  masonry  extending  ten  feet 
below  the  surface  of  the  ground. 

Figs.  26,  27,  28,  and  29  show  different  forms  of  the  field- 
transit  used  by  the  coast  and  other  government  surveys. 
Fig.  26  is  a  very  common  form.  The  telescope  is  26  inches 
focal  length  and  2  inches  aperture.  It  is  provided  with  a 
diagonal  eye-piece  for  observing  transits  of  stars  near  the 
zenith,  the  magnifying  power  being  about  40  diameters.  As 
may  be  seen  from  the  figure,  the  frame  folds  up  so  that  the 
entire 'instrument  may  be  packed  in  a  single  box  of  compara- 
tively small  dimensions.  The  frame  rests  on  three  foot- 
screws  by  means  of  which  it  is  levelled,  the  final  adjustment 
in  this  direction  being  made  by  a  fine  screw  at  the  right  end 
of  the  axis,  as  shown  in  the  figure.  At  the  opposite  end  is  a 
screw,  or  pair  of  screws  acting  against  each  other,  by  means 
of  which  the  final  adjustment  in  azimuth  is  made.  The  two 
lamps  at  opposite  ends  of  the  axis  are  for  illuminating  the 
field.  The  axis  being  perforated,  the  light  enters  it,  falling 
on  a  small  mirror  at  the  intersection  with  the  telescope,  by 
which  it  is  reflected  down  the  tube  to  the  eye-piece.  The 
threads  of  the  reticule  then  appear  as  dark  lines  in  a  bright 
field.  With  some  instruments  there  is  only  one  lamp:  with 
two  the  unequal  heating  and  consequent  expansion  of  the 


PRACTICAL  ASTRONOMY. 


§  159- 


FIG.  27. 


§  l6o.  THE    TRANSIT  INSTRUMENT.  2/1 

two  pivots  is  to  a  great  extent  avoided,  also  the  inconvenience 
of  changing"  the  lamp  from  one  side  to  the  other  when  the 
instrument  is  reversed. 

The  two  small  circles  attached  to  the  telescope  below  the  ' 
axis  are  called  finding-circles ;  they  are  used  for  setting  the 
telescope  at  the  proper  elevation.  They  are  about  6  inches 
in  diameter.  The  alidade  carries  a  level,  as  shown  in  the 
figure.  The  index  is  generally  adjusted  so  as  to  read  zero 
when  the  telescope  is  horizontal.  If  then  the  vernier  is  set 
at  the  meridian  altitude  of  a  star  and  the  telescope  revolved 
until  the  bubble  stands  in  the  middle  of  the  tube,  the  star 
will  be  seen  in  the  middle  of  the  field  when  it  passes  the 
meridian.  One  circle  could  be  made  to  answer  every  pur- 
pose, but  it  would  read  differently  in  the  two  positions  of 
the  axis,  and  this  would  be  likely  to  prove  a  fruitful  source 
of  annoyance.  The  instrument  is  reversed  by  lifting  the 
axis  up  out  of  the  supports  by  hand,  turning  it  around  and 
carefully  replacing  it. 

159.  Fig.  27  shows  a  larger  and  more  complete  instru- 
ment designed  for  longitude  work.     The  focal  length  of  the 
telescope   is   46   inches,    aperture    2f   inches.      Magnifying 
powers  varying   from   80   to    120   diameters   are   used.     A 
special  apparatus  is  provided  for  reversing  the  instrument, 
which  will  be  understood  by  reference  to  the  figure.     The 
cam  worked  by  the   crank  below  the  frame  raises  the  axis 
out  of  its  supports,  when  it  is  turned  around  and  again  low- 
ered into  its  place.     One  of  the  finders  has  two  levels  at- 
tached, one  the  ordinary  finding-level,  the  other  a  much  finer 
one  for  use  in  determining   latitude,  as  will   be  explained 
hereafter. 

160.  Fig.  28  is  a  somewhat  common  form  of  transit,  one 
end  of  the  axis  being  made  to  take  the  place  of  the  lower 
half  of  the  telescope.     A  reflecting  prism  is  placed  at  the 
intersection  of  the  telescope  with  the  axis,  which  bends  the 


272 


PRACTICAL   ASTRONOMY. 


§160. 


FIG.  28. 


l6l.  7W£    TRANSIT  INSTRUMENT.  2?$ 


rays  of  light  at  an  angle  of  90°,  the  eye-piece  being  at  the 
end  of  the  axis. 

The  instrument  shown  in  the  figure  may  be  used  as  a 
transit,  zenith  telescope,  or  azimuth  instrument,  and  is  very 
convenient  for  use  in  positions  where  it  is  not  practicable  to 
have  two  or  three  separate  instruments.  It  has,  besides,  the 
advantage  that,  lor  stars  of  all  zenith  distances,  the  observer 
occupies  the  same  position:  with  the^  common  form  of  instru- 
ment the  position  of  the  observer  is  sometimes  uncomfort- 
able, which  is  prejudicial  to  accuracy. 

161.  Fig.  29  shows  another  form  of  instrument,  made  for 
the  Coast  Survey  by  Fauth  &  Co.  of  Washington.  This 
form  was  first  proposed  by  Steinheil  (Astronomische  Nach- 
richten,  vol.  xxix.  page  177).  Here  a  separate  tube  for  the 
telescope  is  dispensed  with  entirely,  the  axis  being  made  to 
serve  this  purpose  by  placing  the  object-glass  at  one  end  and 
the  eye-piece  at  the  other.  The  reflecting  prism  is  placed 
in  front  of  the  objective,  as  shown  in  the  figure,  and  almost 
in  contact  with  it.  The  tube  is  placed  horizontally  and  in 
the  prime  vertical.  When  the  reflecting  surface  of  the  prism 
is  adjusted  at  the  proper  angle,  the  image  of  any  star  may  be 
made  to  transit  across  the  threads  of  the  reticule,  precisely 
as  in  the  other  forms  of  instruments. 

The  instrument  shown  in  the  figure  has  a  focal  length  of  25 
inches,  and  2  inches  aperture.  It  is  fitted  with  the  appliances 
necessary  to  adapt  it  to  use  as  a  zenith  telescope.  It  is  very 
compact  and  portable,  and  is  therefore  particularly  adapted 
for  use  in  a  rough  country  where  transportation  is  difficult. 

The  portable  transit  instrument  is  mounted  when  practi- 
cable on  a  pier  of  brick  or  stone,  set  into  the  ground  deep 
enough  to  insure  stability.  Where  such  a  foundation  is  not 
available  a  log  sawed  off  square  and  firmly  planted  in  the 
ground  answers  a  very  good  purpose.  The  observatory  may 
be  a  shed  made  of  boards  or  a  canvas  tent. 


274 


PRACTICAL  ASTRONOMY. 


161. 


i63. 


THE    TRANSIT  INSTRUMENT. 


2/5 


•  The  Reticule. 

162.  This  consists  of  a  number  of  spider-lines  arranged  as 
shown  in  the  figure.  The  middle  line  is 
placed  as  nearly  as  may  be  so  that  a  line 
joining  it  with  the  optical  centre  of  the 
object-glass  shall  be  perpendicular  to  the 
axis. 

In  field-instruments  a  very  thin  piece 
of  glass  ruled  with  fine  lines  is  often  used, 
and  is  found  more  satisfactory  in  some  FIG.  30. 

respects  than  the  spider-threads.  In  the  larger  instruments 
intended  to  be  used  with  the  chronograph  there  are  some- 
times as  many  as  twenty-five  lines ;  in  the  smaller  instruments 
there  are  usually  five  or  seven — always  an  odd  number.  The 
two  horizontal  lines  are  for  marking  the  centre  of  the  field. 
The  instrument  should  always  be  set  so  that  the  star  will 
pass  across  the  field  midway  between  them. 


The  Level. 

163.  Every  transit  instrument  is  provided  with  a  delicate 
striding-level.  It  is  supported  by  two  legs,  the  bottoms  of 
which  are  V-shaped.  The  length  is  such  that  these  V's  rest 
on  the  pivots  of  the  axis  when  the  level  is  placed  in  the  posi- 
tion shown  in  Figs.  27,  28,  and  29.  The  tube — which  is 
nearly  filled  with  alcohol  or  sulphuric  ether — is  apparently 
cylindrical,  but  in  reality  has  a  curvature  of  large  radius. 
The  bubble  of  air  which  is  allowed  to  remain  in  the  tube  will 
always  occupy  the  highest  point,  and  so  any  change  in  the 
relative  elevation  of  the  two  ends  will  cause  a  change  in  the 
position  of  the  bubble.  It  may  therefore  be  used  not  only 
for  determining  when  the  axis  is  horizontal,  but,  by  ascertain- 
ing the  angle  corresponding  to  a  motion  over  one  division  of 


2/6  PRAC7"ICAL  ASTRONOMY.  §  164. 

the  graduated  scale,  we  may  by  reading  the  t.wo  ends  of  the 
bubble  determine  the  small  outstanding  deviation  from  per- 
fect adjustment.  The  level  when  so  us'ed  is  a  very  delicate 
instrument  for  angular  measurement. 

164.  To  find  the  value  of  one  division  of  the  level.  This  is 
most  easily  accomplished  by  the  use  of  a  little  instrument 
called  a  level-trier,  which  is  simply  a  bar  of  wood  one  end  of 
which  rests  on  two  pivots,  while  the  other  is  supported  by  a 
micrometer-screw. 

Let  d  =  thp  distance  between  two  consecutive  threads  of 

the  screw  ; 

L  =  the  length  of  the  bar  between  the  points  of  sup- 
port ; 

r  =  the   angle   corresponding  to  one   revolution   of 
the  screw. 

Then  r  =  .    d    „.  .  (270) 

L  sin  i 

Suppose  the  scale  of  the  level  to  read  from  the  middle  in 
both  directions.  Call  the  two  ends  of  the  level  E.  and  W. 
The  readings  in  the  direction  W.  may  be  considered  -f- ; 
those  in  the  direction  E.,  — .  Let  the  level  be  placed  on  the 
bar  of  the  trier,  and  both  ends  of  the  bubble  read ;  then  let 
the  micrometer-screw  be  turned  so  as  to  cause  the  bubble  to 
move  from  its  first  position,  and  the  two  ends  read  again. 

Let  e  and  w  be  the  readings  of  the  bubble  in  the  first 

position  ; 
e'  and  w'  be  the  readings  of  the  bubble  in  the  second 

position ; 

d,  the  value  of  one  division  of  the  level ; 
v,  the  true  angle   through  which  the  bar  has 
been  moved,  as  given  by  the  micrometer- 
screw. 


164. 


VALUE   OF   ONE   DIVISION  OF  LEVEL. 


277 


Then   \(w  —  e)  will  be  the  reading  for  the    middle  of   the 

bubble  in  the  first  position ; 

i(z£/  _  e'}  will  be  the  reading  for   the  middle  Of  the 
bubble  in  the  second  position. 


V  =  -[(a,'  -  /)  -(w-  ,)]  ; 


from  which 


2V 


(w'  —  e')  —  (w  —  e) 


-^ (270 


The  operation  should  be  repeated  many  times  in  different 
parts  of  the  tube  to  insure  greater  accuracy  in  the  final  re- 
sult, and  to  test  the  tube  for  irregularities. 

The  following  example  of  determining  the  value  of  one 
division  of  a  level  is  given  by  Schott,  of  the  Coast  Survey  ; 
for  brevity  only  one  half  of  the  series  is  given  here : 

Coast  Survey  Office,  December  8,  1868.  Determination  of  value  of  one  division 
of  level  B,  belonging  to  Transit  No.  6.  Value  of  one  division  of  level-trier 
=  o".gg. 


Level  B. 

Level- 
trier. 

Change  for 
10  divisions 
of  Trier. 

Tempera- 
ture. 

W. 

E. 

i2h  39m 

210 

91-5 

9.0 

62'.5 

10.75 

220 

80.5 

19-5 

12.  OO 

230 

68.5 

31-5 

11.25 

240 

57-0 

42.5 

9-25 

250 

47-5 

51-5 

I 

9.OO 

260 

38.5 

60.5 

9-25 

270 

29.0 

69.5 

8-75 

280 

20  0 

78.0 

8-75 

52hI2m 

290 

II.  O 

86.5 

62".5 

278  PRACTICAL  ASTRONOMY.  §  165. 

The  numbers  in  the  last  column  but  one  show  that  the 
level  is  not  uniform,  but  there  appears  to  be  a  gradual  change 
of  curvature  from  one  end  towards  the  other.  With  such  a 
level  the  extreme  divisions  ought  never  to  be  used.  If  we 
take  the  mean  of  the  quantities  in  this  column  we  find 

10  divisions  of  level-trier  =  g".g  =  9.875   divisions  of  level. 
Therefore  i  division  =  i".oc>3. 

The  determination  should  be  repeated  at  different  tempera- 
tures to  ascertain  whether  change  of  temperature  affects  the 
curvature  of  the  tube. 

All  fine  levels  are  furnished  with  an  air-chamber  for 
regulating  the  length  of  the  bubble.  When  using  the  level 
this  should  be  kept  at  about  the  length  which  it  had  when 
the  value  of  the  scale  was  being  determined. 

The  value  of  the  level  may  also  be  determined  by  placing 
it  on  a  finely-graduated  circle  and  reading  the  circle  with  the 
bubble  in  different  parts  of  the  tube.  Thus  by  means  of  the 
mural  circle  of  the  Washington  observatory  I  found  the 
value  of  one  division  of  the  level  of  a  zenith  telescope  to  be 
i".o59,  with  a  probable  error  of  o".oi8. 

165.  Adjustment  of  the  Level  of  the  Transit  Instrument. 

The  level  is  used  for  testing  the  horizontality  of  the  axis ; 
therefore  when  it  is  placed  on  the  axis  the  tube  should  be 
parallel  to  the  latter.  If  such  is  the  case — 

First.  The  bubble  must  be  in  the  middle  of  the  tube  when  the 
axis  is  horizontal.  Place  the  level  on  the  axis,  and  bring  the 
latter  approximately  horizontal,  read  the  scale,  reverse  the 
level  and  again  read  the  scale.  If  this  adjustment  is  perfect 
the  reading  will  be  the  same  in  both  positions,  otherwise  one 
half  the  difference  of  the  two  readings  must  be  corrected  by 
raising  or  lowering  one  end  of  the  tube.  The  screws  for  this 
purpose  are  shown  on  the  right  in  Fig.  27.  Repeat  the  pro- 
cess  until  the  adjustment  is  satisfactory. 


§  167.  ADJUSTMENT  OF   THE  INSTRUMENT.  279 

Second.  The  vertical  plane  passed  through  the  axis  must  be* 
parallel  to  that  passed  through  the  tube.  Let  the  level  be  re- 
volved or  rocked  in  both  directions  around  the  pivots  of  the 
axis.  If  the  reading  changes  in  consequence  of  this  motion 
the  adjustment  is  not  perfect.  The  direction  in  which  the 
adjusting-screws  must  be  moved  will  readily  appear  from  the 
motion  of  the  bubble.  The  first  adjustment  should  after- 
wards be  examined,  as  it  may  have  been  disturbed  by  this 
operation. 

Adjustment  of  the  Instrument. 

1 66.  First.     The  threads  of  the  reticule  must  be  in  the  common 
focus  of  the  object-glass  and  eye-piece.    First  adj  ust  the  eye-piece 
by  sliding  it  in  and  out  of  the  tube  until  the  position  is  found 
where  the  threads  are  most  distinctly  seen.     (A  mark  should 
then  be  made  on  the  tube  of  the  eye-piece  so  that  it  may  be 
at  once  set  to  the  proper  focus,  or  a  collar  may  be  fitted  to  it 
so  that  when  it  is  pushed  "  home"  it  will  be  in  focus.)     The 
instrument  should  then  be  turned  to  a  distant  terrestrial  ob- 
ject, or  a  star,  and  the  tube  carrying  the  threads  set  so  that 
the  image  will  remain  constantly  on  one  of  the  threads  when 
the  eye  is  moved  to  one  side  or  the  other  of  the  eye-piece. 
In  some  small  instruments  the  threads  are  fixed  at  the  princi- 
pal focus  of  the  objective  by  the  maker,  with  no  provision  for 
further  adjustment. 

167.  Second.     The  threads  must  be  parallel  to  a  plane  perpen- 
dicular to  the  axis  of  the  instrument.     Direct  the  telescope  to 
a  distant  well-defined  point,  and  bisect  it  with  the  middle 
thread ;  move  the  telescope  up  and  down  through  a  small 
angle   (the  axis   having   been  previously  levelled).     If  the 
thread  is  vertical  it  will  bisect  the  object  throughout  its  en- 
tire extent. 

With  some  instruments  there  is  an  arrangement  for  revolv- 


280  PRACTICAL  ASTRONOMY.  §  1 68. 

ing  the  reticule  and  consequently  for  perfecting  this  adjust- 
ment; with  others  there  is  none.  In  any  case  care  should 
be  taken  to  observe  all  transits  over  the  same  part  of  the  field 
when  a  small  deviation  from  true  verticality  will  not  be  a 
source  of  error. 

1 68.  Third.  To  adjust  the  line  of  collimation. .  Direct  the 
telescope  to  a  distant  terrestrial  point,  and  bisect  it  with  the 
middle  thread  ;  then  carefully  reverse  the  telescope,  and  if 
the  thread  does  not  then  bisect  the  object,  bring  it  half  way 
by  means  of  the  adjusting-screws  found  on  each  side  of  the 
tube  which  contains  the  reticule.  The  operation  must  be 
repeated  until  the  adjustment  is  satisfactory. 

Instead  of  a  distant  terrestrial  point  various  instrumental 
devices  have  been  used,  particularly  in  fixed  observatories. 
One  of  these  is  the  collimating  telescope,  or  collimator  as  it  is 
called.  This  is  a  small  telescope  placed  north  or  south  of 
the  transit  instrument,  so  that  when  the  telescope  of  the  latter 
is  horizontal  the  observer  may  look  through  the  eye-piece 
into  the  object  glass  of  the  collimator.  A  thread  in  the  prin- 
cipal focus  of  the  latter  will  then  appear  precisely  as  if  seen 
from  an  infinite  distance,  since  the  rays  of  light  coming  from 
the  thread  through  the  object-glass  will  all  emerge  in  parallel 
lines.  A  sharply-defined  image  of  this  thread  will  therefore 
be  found  at  the  principal  focus  of  the  transit  telescope,  and 
as  the  thread  itself  is  only  a  few  feet  distant,  this  image  will 
not  be  disturbed  by  atmospheric  undulations  as  in  the  case 
of  a  distant  mark.  By  using  two  collimators,  one  north  and 
one  south,  the  adjustment  may  be  made  without  reversing 
the  instrument ;  this  process,  however,  cannot  be  conven- 
iently applied  to  a  field-instrument. 

The  mercury  collimator  is  also  much  used  with  the  fixed 
instruments  of  observatories.  This  is  simply  a  basin  of 
mercury  placed  directly  under  the  telescope,  so  that  when 
tne  latter  is  placed  vertical  with  the  objective  down  the 


§169.  ADJUSTMENT  IN    THE   MERIDIAN.  28 1 

observer  can  look  through  the  eye-piece  into  the  mercury. 
The  threads  will  then  be  seen  in  the  field,  together  with  their 
images  reflected  from  the  mercury.  The  axis  having  been 
carefully  levelled,  the  thread  and  its  reflected  image  will  co- 
incide if  there  is  no  error  of  collimation.  If  the  collimation 
has  been  previously  adjusted  by  the  collimating  telescope, 
this  process  may  be  employed  for  measuring  the  inclination 
of  the  axis;  it  is  not,  however,  a  suitable  method  to  employ 
with  the  portable  instrument. 

169.  Fourth.  To  adjust  the  instrument  in  the  plane  of  the 
meridian.  The  transit  is  used  in  connection  with  the  sidereal 
chronometer.  The  observations  will  be  made  for  determining 
the  error  of  the  chronometer;  this  is,  therefore,  presumably 
not  known  with  any  degree  of  accuracy. 

If  nothing1  whatever  is  known  of  the  chronometer  error, 
it  may  in  certain  cases  be  advisable  to  determine  it  approxi- 
mately by  the  sextant,  or  by  the  altitude  of  a  star  measured 
with  the  vertical  circle  of  an  engineer's  theodolite.  Such  a 
preliminary  determination  will  very  seldom  be  necessary. 

As  the  approximate  time  may  therefore  be  known  by  some 
process,  we  first  take  the  best  value  available.  Suppose,  for 
simplicity,  the  chronometer  to  be  set  for  this  approximate 
time — or,  in  other  words,  that  to  the  best  of  our  knowledge 
the  time  shown  by  the  chronometer  is  correct.  We  then 
take  from  the  Nautical  Almanac  the  right  ascension  of  a  close 
circumpolar  star,  and  as  this  is  equal  to  the  sidereal  time  of 
culmination,  we  direct  the  telescope  to  the  star,  level  the  axis, 
and  at  the  instant  when  the  time  shown  by  the  chronometer 
equals  this  right  ascension  bring  the  middle  thread  of  the 
reticule  on  the  star,  using  the  fine-motion  screw  at  the  end 
of  the  axis  for  the  final  adjustment.  The  instrument  will  now 
be  approximately  in  the  meridian.  We  next  level  the  instru- 
ment carefully  by  the  fine-motion  screw  at  the  end  of  the 
axis,  and  select  "from  the  almanac  a  star  which  culminates 


282  PRACTICAL  ASTRONOMY.  §  169. 

near  the  zenith  for  determining  a  more  correct  value  of  the 
time,  or  of  the  chronometer  correction.  As  all  vertical  cir- 
cles pass  through  the  zenith,  by  selecting  a  star  which  passes 
as  near  as  possible  to  this  point  we  determine  a  very  close 
approximation  to  the  true  chronometer  correction,  even 
when  the  instrument  has  a  large  azimuth  error.  It  is  better 
to  use  two  stars  for  this  purpose,  one  culminating  north  of 
the  zenith,  and  one  south  (as  it  will  very  seldom  be  possible 
to  find  a  star  culminating  exactly  in  the  zenith).  If  the 
operations  already  described  have  been  carefully  attended 
to  we  shall  now  know  our  chronometer  correction  within  a 
second,  which  will  be  accurate  enough  for  perfecting  the 
adjustment  in  the  meridian  by  another  circumpolar  star. 

Let  A®  =  the  value  of  the  chronometer   correction  just 

determined ; 
a  =  the  right  ascension  of  any  star. 

Then    a  —  dQ  —  the  chronometer  time  of  culmination. 

When  the  chronometer  indicates  this  time,  the  star  must  be 
carefully  bisected  by  the  middle  thread,  the  axis  having  been 
previously  levelled.  If  the  observer  does  not  yet  feel  suffi- 
cient confidence  in  the  adjustment,  the  operation  must  be 
repeated  for  a  closer  approximation. 

The  circumpolar  stars  most  suitable  for  this  adjustment 
are  the  four  standard  stars  of  the  Nautical  Almanac,  viz., 
a,  d,  and  A,  Ursas  Minoris  and  51  Cephei.  Besides  these  the 
ephemeris  for  1885  and  following  years  gives  a  number  of 
other  stars  near  the  pole  reduced  to  apparent  place  for  inter- 
vals of  ten  days. 


§  1 70.  METHODS  OF  OBSERVING.  283 

Methods  of  Observing. 

170.  The  immediate  aim  of  the  observer  is  to  obtain  as 
accurately  as  possible  the  instant  of  time,  as  shown  by  the 
clock  or  chronometer,  when  the  star  crosses  each  thread  of 
the  reticule.  These  times  may  then  be  reduced  by  a  method 
to  be  explained  hereafter  to  the  time  over  the  middle  thread. 
If  then  r  is  the  probable  error  of  a  transit  observed  over  a 
single  thread,  and  n  the  number  of  threads  observed,  the 

probable  error  of  the  mean  will  be  — -. 

\/n 

There  are  two  methods  of  observing-  transits,  viz.,  the  eye 
and  ear  method  and  the  chronographic  method.  The  latter 
method  is  more  accurate  except  with  an  observer  of  long 
experience,  an^i  is  now  used  almost  universally  in  fixed  obser- 
vatories. It  is  also  employed  in  the  field  when  the  time  is 
required  with  great  accuracy  for  longitude  work. 

In  other  cases,  when  the  portable  instrument  is  used,  the 
observations  will  be  made  by  the  eye  and  ear  method,  which 
is  as  follows:  A  few  seconds  before  the  star  to  be  observed 
reaches  the  thread  the  observer  takes  the  time  from  the 
chronometer  and  watches  the  star  as  it  approaches  the  thread, 
at  the  same  time  counting  .the  beats  of  the  chronometer. 
When  the  star  crosses  the  thread  the  exact  instant  is  noted  ;  if 
the  thread  is  crossed  between  two  beats, 
the  fractional  part  of  a  second  is  esti- 
mated to  the  nearest  tenth.  This  esti- 
mation is  made  more  by  the  eye  than  ~  a. 

the  ear;  thus,  suppose  when  the  obser-  

ver  counts  ios  the  star  is  at  a,  and  when 

ii3  at  b\  the  distance  from   a  to  the 

thread  will  be  compared  with  the  dis-  FIG  32. 

tance  from  a  to  b,  and  the  ratio  will  be  expressed  in  tenths.    In 

this  case  the  time  will  be  ios.4.    A  skilful  observer  will  seldom 


2  84  PR  A  CTICA  L   AS  TRONOM  Y.  '  §  1 7  I . 

be  in  error  by  so  much  as  T2T  of  a  second  in  estimating  the 
time  over  a  single  thread  for  a  star  near  the  equator. 

By  the  chronographic  method  the  observer  registers  the 
instant  when  the  star  is  on  the  thread  by  simply  pressing  the 
key  which  closes  or  breaks,  as  the  case  may  be,  the  galvanic 
circuit.*  This  instant  is  recorded  by  a  mark  on  the  cylinder 
of  the  chronograph,  and  may  be  read  off  at  leisure.  As  the 
observer  is  not  obliged  to  count  the  seconds  as  in  the  other 
method,  the  threads  may  be  placed  much  closer  together 
and  a  larger  number  of  readings  taken.  A  practical  limit 
will,  however,  soon  be  reached  beyond  which  nothing  will 
be  gained  in  accuracy  by  increasing  the  number  of  threads. 

Formerly  the  large  transits  of  the  Coast  Survey  were  pro- 
vided with  twenty-five  threads  arranged  in  five  groups,  or 
tallies  of  five  threads  each.  Of  late  this  number  has  been  re- 
duced to  thirteen,  the  central  tally  containing  five  threads,  the 
two  on  each  side  three  each,  and  the  two  extreme  tallies  only 
one  each.  The  middle  threads  of  the  tallies  are  at  equal  dis- 
tances and  may  be  used  for  eye  and  ear  observation,  while  the 
middle  tally  is  convenient  for  observing  close  circumpolar 
stars,  which  may  be  best  observed  by  the  eye  and  ear  method. 

Mathematical  Theory  of  the  Transit  Instrument. 

171.  We  have  shown  how  to  adjust  the  instrument  and 
place  it  in  the  plane  of  the  meridian.  With  whatever  care 
these  adjustments  are  made,  there  will  always  remain  small 
outstanding  errors,  the  existence  of  which  will  affect  the 
observed  time  of  a  star's  transit.  The  amount  of  these  errors 
must  then  be  determined,  and  the  necessary  corrections 
applied  to  the  observed  time  to  reduce  it  to  the  true  time  of 
meridian  passage. 

*See  Art.  121. 


§171-          THEORY   OF   THE    TRANSIT  INSTRUMENT.  285 

We  shall  call  a  line  passing  through  the  centres  of  the 
pivots  and  produced  indefinitely  the  rotation  axis.  x^lso, 
the  line  drawn  through  the  optical  centre  of  the  object-glass 
and  perpendicular  to  the  rotation  axis  is  the  collimation  axis. 
When  the  instrument  is  revolved  this  line  describes  a  great 
circle  of  the  celestial  sphere,  the.poles  of  which  are  the  points 
where  the  rotation  axis  pierces  the  sphere.  When  these 
poles  are  known  the  position  of  the  circle  itself  is  known. 

Let    90°  —  a  =  the  azimuth  of  the  point  where  the  west 

end  of  the  axis  pierces  the  sphere ; 
b  —  the  altitude  of  the  same  poinii. 

Then  a  will  be  the  deviation  of  the  axis  from  the  true  east  and 
west  position,  plus  when  the  west  end  deviates  to  the  south  , 
and  b  is  the  deviation  from  the  true  horizontal  position,  plus 
when  the  west  end  is  high. 

Let     90°    -  m  =  the  hour-angle  of  this  point; 

n  —  the  declination. 

Let  x,  y,  2  be  the  rectangular  co-ordinates  of  this  point 
referred  to  the  horizon. 

4 

Then  A,  90°  —  a,  and  b  will  be  the  polar  co-ordinates,  and 
we  have  * 

x  =  A  cos  b  cos  (90°  •-  a)  =  A  cos  b  sin  a ;  \ 

y  —  A  cos  b  sin  (90°  •  -  a)  —   4  cos  b  cos  a  ;  >     (272) 

z  =  A  sin   b.  ) 

Let  xr ,y' ,  z'  be  the  rectangular  co-ordinates  referred  to  the 
equator. 

*See  equations  (no). 


286  PRACTICAL   ASTRONOMY.  §  I/ 1. 

Then  J,  (90°  —  m),  and  n  are  the  polar  co-ordinates,  and 

x'  =  A  cos  n  cos  (90°  —  m)  =.  A  cos  n  sin  #2 ;  j 

y'  —  A  cos  #  sin  (90°    -  m)  —  ^/  cos  «  cos  w ;  >  (273) 

z'  =  A  sin  n.  ) 

The  formulae  for  transformation  of  co-ordinates  will  be  * 

x'  —  x  sin  <p  -\-  z  cos  <?> ;       ) 

/  =/;  [.  .    .    .    (274) 

<sr'  =  —  ;tr  cos  (p  -\-  z  sin  cp.  } 

Substituting  for  x,  y,  z  and  *',  y,  s'  their  values,  and  dropping 
the  common  factor  A,  we  have 

cos  n  sin  ;«  =  cos  b  sin  #  sin  q>  -f-  sin  ^  cos  (p ;        i 
cos  n  cos  /«  =  cos  b  cos  #  ;  >•  (275) 

sin  n  —  —  cos  b  sin  #  cos  (p  +  sin  £  sin  <£>.  ) 

Equations  (275)  give  m  and  #  when  #  and  b  are  known. 
No  limit  has  been  placed  to  the  values  of  a,  b,  m,  and  n,  which 
may  therefore  be  of  any  magnitude,  and  consequently  the 
instrument  in  any  position.  By  careful  adjustment,  how- 
ever, these  quantities  may  always  be  made  very  small,  and 
there  will  therefore  be  no  appreciable  error  in  writing  the 
quantities  themselves  for  their  sines,  and  writing  for  the 
cosines  unity.  Therefore 

For  the  transit  instrument  in  the  meridian, 

m  —  a  sin  q>  +  b  cos  tp ;      ]  ,     $, 

n  =  —  a  cos  y>  -f-  b  sin  q>.  \  ' 

From  these  we  readily  derive 

a  —  m  sin  cp  —  n  cos  (p  ;  )   ^  /      x 

b  =  m  cos  (p  +  n  sin  (p.    } 

*See  equations  (112.) 


§  1/2.          THEORY   OF   THE    TRANSIT  INSTRUMENT.  287 

172.  Now  let  r  =  the  east  hour-angle  of  a  star  when  seen 

on  the  middle  thread  ; 

c  =  the  error  of  collimation ;  plus  when  the 
star  reaches  the  thread  too  soon.* 

Now  let  the  star  when  on  the  middle  thread  be  referred  to 
a  system  of  rectangular  co-ordinates,  the  plane  of  x,  y  being 
the  plane  of  the  equator,  the  axis  of  x  being  perpendicular 
to  the  rotation  axis. 

Then  8  —  the  star's  declination  is  the  angle  formed  with 

the  plane  of  x,  y,  by  the  radius  vector ; 
T ;.  —  m  =  the  angle  formed  with  the  axis  of  x  by  the  pro- 
jection of  the  radius  vector  on  the  plane  of  x,  j. 

Then  x  =  A  cos  #  cos  (r  —  m) ;  ) 

y  —  A  cos  d  sin  (r  —  m] ;  I .    .     .     .     (278) 
z  =  A  sin  $ ;  ) 

y  being  reckoned  towards  the  east. 

Let  the  star  be  now  referred  to  a  new  system  of  co-ordi- 
nates in  which  the  axis  of  x  coincides  with  that  of  the  last  sys- 
tem, the  axis  of  y  being  the  rotation  axis  of  the  instrument. 

Then  c  =  the  angle  formed  with  the  plane  of  x,  z,  by  the 

radius  vector ; 

&i  =  the  angle  formed  with  the  axis  of  x  by  the  pro- 
jection of  the  radius  vector  on  the  plane  of  x,  z. 

Then  x'  =  A  cos  c  cos  d,  ;  \ 

y'  =  Jsin^;  V (279) 

z1  =  A  cos  c  sin  tf,.   ) 

*  The  star  is  supposed  to  be  observed  at  upper  culmination. 


288  PRACTICAL   ASTRONOMY.  §  172. 

In  these  two  systems  the  axes  of  x  coincide,  tne  axes  of 
y'  and  z'  make  the  angle  n  with  those  of  y  and  z.     Therefore 


y'  —  y  cos  n  —  2  sin  «;>.-.     .     .     .     (280) 
z'  •=.  y  sin  n  -\-  z  cos  n.  ) 

i 
Combining  (278),  (279),  and  (280),  we  have 

cos  c  cos  tfj  =  cos  d  cos  (r  —  m);  ) 

sin  c  =  cos  &  sin  (r  —  ;#)  cos  n  —  sin  tf  sin  n;  >     (281) 
cos  c  sin  tf,  =  cos  d  sin  (r  —  ;/z)  sin  n  +  sin  #  cos  n.  } 

With  these  equations,  as  with  (275),  no  restrictions  have 
been  placed  on  the  quantities  involved,  and  they  will  serve 
for  computing  r  when  m,  n,  and  c  are  known.  When  these 
quantities  are  small,  as  with  the  instrument  adjusted  in  the 
meridian,  the  second  of  (281)  becomes 

c  =  (r  —  m)  cos  d  —  n  sin  d; 
from  which        r  =  m  +  //  tan  3  -f~  c  sec  #• .    •     •     •     •     (282) 

This  is  BesseV s  formula  for  computing  the  hour-angle  of  the 
star  when  it  passes  the  middle  thread  of  the  reticule.  In  ap- 
plying it,  the  unit  in  which  m,  n,  and  c  are  expressed  must  be 
the  second  of  time. 

If  we  substitute  in  (282)  the  value  of  m  from  the  second  of 
(277),  viz., 

m  =  b  sec  (p  —  n  tan  g>, 

« 
we  have     r  =  b  sec  cp  -f-  n  (tan  d  —  tan  <p)  -f-  c  sec  3.    (283) 

This  is  Hanseris  formula  for  computing  r.  We  see  from  it 
that  when  d  =  gj,  the  term  in  n  vanishes  and  r  depends  on  b. 
and  c  alone.  From  this  it  follows  that  those  stars  are  best 


§173-  CORRECTION  FOR  DIURNAL   ABERRATION.  289 

suited  for  determining  r  —  and  therefore  the  clock  correction  — 
which  culminate  near  the  zenith. 

Substituting  in  Bessel's  formula  the  values  of  m  and  n 
from  (276),  we  readily  find 

sin    c    —  d  cos     >  —  6  c 


Which  is  Mayer'  s  formula,  and  is  the  one  best  adapted  for 
use  with  the  portable  transit. 

We  adapt  these  formulae  to  the  case  of  lower  culmination 
by  changing  6  into  180°  —  tf. 

Now  let  a  =  the  apparent  right  ascension  of  any  star; 

&  =  the  observed  clock  time  of  the  stars  passing 

the  middle  thread; 
A®  =  the  clock  correction. 


Then  a=9  +  Je  +  T.,) 

A®  =  «--(©+  T).  j  V 

In  which  r  may  be  computed  by  either  (282),  (283),  or  (284). 
If  the  star  is  observed  at  lower  culmination,  a  becomes 
I2h  +  a. 

Correction  for  Diurnal  Aberration. 

173.  Aberration  is  the  apparent  change  in  a  star's  posi- 
tion caused  by  the  progressive  motion  of  light  combined 
with  the  motion  of  the  earth  itself.  The  displacement  is  in 
the  direction  of  the  earth's  motion,  and  the  tangent  of  the 
angle  of  displacement  is  equal  to  the  component  of  the  veloc- 
ity of  the  earth  perpendicular  to  the  line  of  sight  divided  by 
the  velocity  of  light. 

Aberration  is  considered  under  two  heads,  viz.,  annual  and 
diurnal  aberration,  the  former  resulting  from  the  earth's  an- 


290  PRACTICAL  ASTRONOMY.  §  173. 

nual  motion  in  its  orbit,  and  the  latter  from  the  revolution  on 
its  axis.  The  subject  will  be  treated  in  a  subsequent  chapter  as 
fully  as  will  be  necessary  for  our  purposes.  At  present  we 
shall  only  consider  the  diurnal  aberration. 

Let  k  =.  the  diurnal  aberration  of  an  equatorial  star  at  the 
time  of  transit.  The  velocity  of  light  is  186,380  miles  per 
second.  A  point  on  the  earth's  equator  has  a  linear  motion 
of  0.2882  mile  per  second,  in  consequence  of  the  diurnal  rev- 
olution of  the  earth.  Therefore  the  linear  velocity  of  a  point 
whose  latitude  is  q>  will  be  0.2882  cos  q>.  Then 


If  the  star's  declination  is  tf,  the  effect  upon  the  star's 
hour-angle  being  £',  we  have,  by  applying  Napier's 
first  rule  for  right-angle  triangles  to  the  triangle  shown 
.  in  the  figure, 


33. 


sin  k  =  sin  k'  cos  #; 
or  k'  —  k  sec  8  =  S.O2  1  cos  q>  sec  d.    .     .     (287) 

As  this  will  cause  the  star  to  appear  too  far  east,  the  ob- 
served time  of  culmination  will  be  too  late  and  the  correc- 
tion must  be  subtracted. 

The  correction  for  diurnal  aberration  may  be  combined 
with  the  collimation  constant  by  making 

cf  =  c  —  6.02i  cos  (p.     ......     (288) 

As  observations  are  made  in  both  positions  of  the  axis,  it 
is  necessary  to  distinguish  between  them.  This  may  be  done 
by  noting  the  position  of  the  clamp,  whether  it  is  east  or  west. 
If  then  the  sign  of  c  is  determined  for  clamp  west,  the  alge- 


§  !74-  EQUATORIAL  INTERVALS   OF   THREADS.  29 1 

braic  sign  must  be  changed  when  the  position  is  clamp  east. 
It  must  be  remembered  that  the  algebraic  sign  of  the  aber- 
ration does  not  change  when  the  instrument  is  reversed;  so  if 
this  correction  has  been  combined  with  c,  c'  will  in  one  case 
be  the  sum  of  the  two,  and  in  the  other  case  the  difference. 


Equatorial  Intervals  of  the  Threads. 

174.  When  the  transit  of  a  star  over  one  of  the  side  threads 
is  observed,  we  may  regard  the  distance  of  this  thread  from 
the  collimation  axis  as  its  error  of  collimation,  and  proceed 
with  the  reduction  precisely  as  in  case  of  the  middle  thread. 
It  is  simpler  in  practice,  however,  to  determine  the  angular 
distances  of  the  side  threads  from  the  middle  thread,  when 
the  times  may  all  be  reduced  to  the  time  over  this  thread. 
This  angular  distance  when  expressed  in  time  is  evidently 
the  time  required  for  an  equatorial  star  to  pass  from  the  side 
thread  to  the  middle  thread. 


Let        i  =  the  equatorial  interval  for  any  thread; 

/  —  the  interval  for  a  star  whose  declination  is  $. 
Then /-[-£  =  the  collimation  error  for  this  thread; 

r  +  /  =  the  hour-angle  of  a  star  when  seen  on  this  thread. 

The  second  of  equations  (281)  may  be  written 

sin  (r  —  m)  =  sin  c  sec  n  sec  3  -\-  tan  n  tan  #, 
and  for  the  side  thread 
sin  (r  -f-  /  —  m)  =  sin(z  -)-  c)  sec  n  sec  d  +  tan  n  tan  tf. 


292  PRACTICAL   ASTRONOMY.  §  174. 

By  subtraction, 

sin  (r  +  /  —  *«)  —  sin  (r  —  /»)  =  [sin(*  +  ^)  —  sin  c\  sec  #  sec  £ ; 

which  becomes 

2  cos  (-J-7  +  r  —  w)  sin  £7  =  2  cos  (%i  +  <:)  sin  £z  sec  n  sec  tf. 

Since  r  —  m  and  #  are  very  small  quantities,  the  above 
may  be  written 

sin  7  =  sin  i  sec  tf (289) 

For  all  stars  not  nearer  the  pole  than  10°, 

I  —  i  sec  $ (289), 

When  7  is  observed  and  i  is  required,  the  equations  become 

sin  i  =  sin  7  cos  #; (290) 

i  —  I  cos  £ (29o)t 

When  the  star  is  nearer  the  pole  than  10°,  formulas  which 
are  practically  exact  are  obtained  as  follows:  /may  always 
be  written  for  sin  i,  and  (7—  £73)  for  sin  7.  Therefore 

i  =  7(i  —  fT)  cos  d. 

But     cos  7  =  i  -  -J72        and        (cos  /)*  =  i  -  ^72; 

therefore  we  have 

i  =?  7  cos  tf  V  cos7. (291) 

/  =  *'  sec  (J  Vsec7. (291), 


§175- 


EQUATORIAL  INTERVALS   OF    THREADS. 


293 


The  following  table  gives  log  V  cos /and  log  V  sec /with 
the  argument  /  in  time : 


7. 

log  ^  cos  7. 

/- 

log  V  cos  7. 

7. 

log  V  sec  7. 

log  V  sec  7. 

log  V  sec  7. 

log  V  cos  7. 

im 

9-99999 

0.00000 

I&» 

9.99965 

0.00035 

3Im 

9.99867 

0.00133 

'2 

99 

OI 

17 

960 

040 

32 

858 

142 

3 

99 

OI 

18 

955 

045 

33 

849 

151 

4 

98 

02 

19 

95° 

050 

34 

840 

160 

5 

97 

°3 

20 

945 

055 

831 

169 

6 

95 

°5 

21 

939 

06  1 

36 

821 

179 

•  7 

93 

07 

22 

933 

067 

37 

811 

189 

8 

09 

23 

927 

073 

38 

800 

200 

9 

89 

ii 

24 

921 

079 

39 

.  789 

211 

10 

86 

14 

25 

914 

086 

40 

778 

222 

ii 

83 

26 

907 

093 

767 

233 

12 

80 

20 

27 

899 

101 

42 

756 

244 

13 

77 

23 

28 

892 

lot 

43 

744 

256 

14 

73 

27 

29 

884 

116 

44 

732 

268 

15 

9.99969 

0.00031 

3° 

9.99876 

0.00124 

45 

9.99719 

0.00281 

175.  Suppose  the  reticule  to  contain  five  threads. 

Let        T  =  the  time  of  a  star's  passing  the  middle  thread; 
A*  t»  /3,  /4,  /B  =  the  times  of  passing  the  separate  threads ; 
*'i»  *»  *B>*  *4>  h  =  tne  equatorial  intervals. 

The  star  is  supposed  to  pass  the  threads  in  the  above  order 
when  the  clamp  is  west.  When  the  position  is  clamp  east,  the 
order  will  be  reversed,  becoming  z*5,  z'4,  iv  iv  ir  At  lower 
culmination  the  order  will  be  the  reverse  of  that  of  upper 
culmination. 

We  shall  have        T  =  t,  +  ^  sec  3 
=  /,  -f-  z*3  sec  d* 
=  t*  +  i»  sec  d. 


*  When  the  reduction  is  to  the  middle  thread,  z'3  =  o. 


294  PRACTICAL  ASTRONOMY.  §  177. 

The  mean  is 

T  =  ^-*.  +  *»+j«±*.  +  L+  *•  + *.  *  - hV+J.  sec  j;  (292) 

or  T'  =  7*0  -f-  ^z  sec  tf  for  clamp  west; 

T  =  T0  —  Ai  sec  #  for  clamp  east. 

Instead  of  reducing  the  observed  times  to  the  time  over 
the  middle  thread,  we  may  reduce  them  to  the  time  over  an 
imaginary  thread,  the  time  over  which  is  the  mean  of  the 
times  over  the  five  threads,  or  T0  of  the  above  formula. 
The  equatorial  intervals  and  error  of  collimation  are  then 
determined  with  reference  to  this  mean  thread  instead  of  the 
middle  thread.  This  method  is  more  convenient  than  the 
preceding,  as  Ai  then  vanishes  and  the  equatorial  intervals 
are  not  required  when  all  of  the  threads  are  observed. 

Reduction  of  Imperfect  Transits. 

176.  A  transit  is  imperfect  when  the  time  over  one  or  more 
of  the  threads  has  not  been  observed.     Formula  (292)  applies 
equally  to  such   a  transit,   by   simply   dropping  the  terms 
corresponding  to  the  threads  which  were  not  observed.    Thus 
suppose  the  first  two  threads  were  not  observed;  the  formula 
will  then  be 

r^i+Aii  +  id^hi.*,.    • 

Correction  for  Rate.  » 

177.  If  the  rate  of  the  chronometer  is  large,  it  may  be 
necessary   to  take  it  into  account  in  reducing   imperfect 
transits. 

*  When  the  reduction  is  to  the  middle  thread,  z'3  =  o. 


§1/8-  DETERMINATION   OF    THE    CONSTANTS.  295 

Let      $T  =  the  hourly  rate  of  the  chronometer. 
Then  if  i  is  given  in  seconds,  we  shall  have 


Thus  if  a  star  is  observed  with  a  mean  time  chronometer, 
6T  =  9S.83O  and  (293)  becomes 

T  =  t  +  i  sec  d  x  0.99727;  \  ,      . 

or  T  =  t  +  i  sec  d  [9.99881].    \  ' 


Determination  of  the  Constants. 

178.  We  may  determine  the  time  of  the  stars  passing  the 
meridian,  and  consequently  the  clock  correction,  from  for- 
mulas (284)  and  (285)  when  we  know  the  values  of  a,  b,  and  c, 
or  from  formulas  (282)  and  (285)  when  we  know  m,  n,  and  c. 
The  determination  of  these  quantities  will  therefore  now 
be  considered. 

The  Level  Constant,  b. 

Place  the  stridirig-level  on  the  axis  and  read  both  ends  of 
the  bubble,  reverse  the  level  and  read  again. 

Let  w  and  e  be  the  readings  of  the  west  and  east  end  in 

first  position; 
w'  and  *',  the  readings  of  the  west  and  east  end  in  sec- 

ond position; 
d,  the  value  of  one  division  of  the  level  expressed  in 

time; 
x,  the  error  of  the  level  due  to  any  want  of  perfect  ad- 

justment. 


296 


PRACTICAL  ASTRONOMY. 


§  179- 


Then  if  there  were  no  error  the  inclination  would  be  equal 
to  the  reading  of  the  middle  point  of  the  bubble,  or 

b  =  \d(w  —  e)  +  x\ 
b  =  %d(w'  -  e'}  -  *; 


the  mean  of  which  is 


(295) 


The  level  is  often  reversed  two  or  more  times  for  greater 
accuracy.  Whatever  the  number  of  reversals,  the  inclination 
is  given  by  the  formula 

d~™    -  £];  .  (296) 


where  W  and  E  are  respectively  the  means  of  the  east  and 
west  readings. 

Inequality  of  Pivots. 

179.  The  above  expression  for  b  is  obtained  by  applying 
the  level  to  the  outer  suri  .ce  of  the  pivots;  it  therefore  gives 
the  true  inclination  of  the  rotation  axis  only  when  the  diam- 
eters of  the  pivots  are  eq  jal.  If  they  are  unequal  this  value 
of  b  requires  a  correction  determined  as  follows: 

Fig  34^  is  a  cross-section  of  one  of  the  pivots,  with  the  V 
of  the  level  B,  and  of  the  instrument  A.  Suppose  the  clamp 


EC 


FIG.  34«.  FIG.  34$. 

west.  Formula  (295)  gives  the  inclination  of  the  line  B'B; 
that  of  ^'^is  required.  Suppose  the  V  of  the  level  to  have 
the  same  angle  as  the  V  of  the  instrument. 


§  179-  INEQUALITY   OF  PIVOTS.  297 

Let  B  and  B'  be  the  inclinations  as  shown  by  the  level  for 

clamp  west  and  east  respectively; 
b  and  b',  the  true  inclinations  of  C'C', 
/?,  the  constant  inclination  of  A'A\ 
p,  the  angle  ECC'  =  C'CF. 

For  clamp  west,      b   —  B  +  /;         b  =  P  —  p\  )        /  N 
For  clamp  east,       b'  =  B'  -  /;        b'  =  /3  -f/.'  j 

By  subtraction,      b'  —  b  —  B'  —  B  —  2p  =  2/; 

B'  -  B 


(297) 


Which  determines  the  value  of/.  In  order  to  be  reliable  it 
must  be  derived  from  a  large  number  of  readings  of  the  level 
in  both  positions  of  the  axis.  It  will  then  be  a  correction  to 
be  added  algebraically  to  the  inclination  as  given  by  the  level 
for  the  position  clamp  west,  or 

b  =  B  +  /  for  clamp  west; ) 
V  —  B'  -  p  for  clamp  east.    )  * 

If  the  angle  of  the  level  V  is  not  equal  to  that  of  instrument  V,  the  angle  E CC1 
will  not  be  equal  to  C'CF  and  we  proceed  as  follows: 

Let  2*  =  the  angle  of  the  level  V; 

2t!  =  the  angle  of  the  instrument  V; 
r  and  r'  =.  the  radii  of  the  pivots; 
d  •=  jBCin  the  figure; 
d\  =  AC  in  the  figure; 
L  =  length  of  level  =  C'C; 
p  -  angle  ECC1  \ 
pi  =  angle  C'CF; 

the  notation  in  other  respects  remaining  as  before. 


298  PRACTICAL  ASTRONOMY.  §  1 79- 

Then  for  end  next  the  clamp  d  =  ^.;          </i  = 


Then  for  end  remote  from  clamp     d '  =  -r—. ; 

sin  i 


d'  —  d        r  —  r 


L  L  sin  i'  L  sin  i  sin  15 

sin/i  —       -£ Zsinti'  ~  Zsin  »i  sin  15' 


Dividing  (</)  by  (r)  we  have  /  =  ihi^ W 

Then  ^  =  ^  +/;        b  =  ft  -  p^ 

b'  —  b  =  B'  —  B  —  2/  =  2/1 ; 

2          —  ^"T/7!- 

Substituting  the  value  of /i  from  (e)  and  reducing,  we  readily  find 


=      —  .-. 

2       \sin  i  -\-  sin  Zi/ 

Example.  The  following  readings  of  the  level  were  made 
for  determining  the  inequalities  of  the  pivots  of  the  transit 
instrument  of  the  Sayre  observatory. 

Clamp  East.  Clamp  West. 

E.  w.  E.  w. 

Direct,          144     l$-1  I2'8     l6'2 

Reversed,  '  12.7     16.7  14-6     14-9 

(,  +  *')  =  27.1     31.8  =  w  +  «/     27.4     31.1 
By  formula  (295),    B'  =  +  i.i;5;  B  =  +  -925; 

.#  and  -5'  being  expressed  in  terms  of  one  division  of  the 
level. 


§i8i. 


INEQUALITY   OF  PIVOTS. 


299 


The  angle  of  the  level  V  was  equal  to  that  of  the  transit; 
therefore,  by  (297), 

B'  -  B 
p  =  -——  =  +  -062. 

By  a  considerable  number  of  readings  made  at  different 
times  the  following  values  of  /  were  obtained.  The  first 
and  third  columns  show  the  angle  of  elevation  of  the  tele- 
scope, the  second  and  fourth  the  corresponding  values  of/. 


o° 

-f.056 

125° 

.042 

IO 

.080 

130 

.059 

20 

.068 

140 

.052 

30 

.056 

150 

.076 

40 

.077 

1  60 

.069 

50 

.046 

170 

.064 

60 

.062 

Mean  of  .3  values/  =  T  .062  | 
The  value  of  one  division  of  the  level  is  d  =  ".174;  therefore  p*  =  ".oil. 

180.  The  diameters  of  the  pivots  may  not  only  be  unequal, 
but  the  forms  may  be  irregular.     This  is  tested  by  reading 
the  level  with  the  telescope  placed  at  different  zenith  dis- 
tances.    If  inequalities  are  found  to  exist,  a  table  of  correc- 
tions for  different  zenith  distances  from  zero  to  90°  on  each 
side  of  the  zenith  may  be  formed  in  case  it  is  necessary  to  use 
the  instrument  in  this  condition.     If  the  corrections  are  large 
enough  to  be  appreciable,  however,  the  instrument  should 
be  put  into  the  hands  of  an  instrument-maker  for  repairs. 

181.  A  little  instrument  designed  by  Prof.  Harkness,  and 
called  by  him  the    "  spherometer-caliper,"  is  very  conven- 
ient for  measuring  the  inequalities  and  irregularities  of  pivots. 

Fig.  35#  is  a  front  and  35^  a  side  elevation.     The  same 


300 


PRACTICAL  ASTRONOMY. 


181. 


letters  refer  to  both  figures.  The  foundation-plate  b  carries 
two  cylindrical  guides,  dd,  which  are  connected  at  their  lower 
end  by  the  bar  e.  Into  the  foundation-plate  is  screwed  the 
brass  piece  m,  to  which  is  cemented  the  thick  circular  glass 
plate  c.  The  two  V's,  aa,  are  also  firmly  screwed  to  the  foun- 
dation-plate. The  brass  plate  /slides  freely  up  and  down 


FIG.  350;. 


FIG.  35$. 


THE  SPHEROMETER-CALIPER. 


between  the  guides  dd,  being  kept  in  place  by  three  loops, 
two  of  which  pass  around  the  right-hand  guide  and  one 
around  the  left,  as  shown  in  the  figure.  The  brass  rod  g, 
which  passes  through  the  piece  m  and  the  plate  c  without 
touching  either  of  them,  is  firmly  attached  to  the  upper  end 
of  the  plate/,  and  moves  with  it,  while  to  the  lower  end  of 


§  l8l.  INEQUALITY  OF  PIVOTS.  3O1 

f  is  attached  a  second  short  brass  rod  which  passes  freely 
through  the  bar  e  and  carries  the  nut  h. 

In  using  the  instrument,  the  plate  /is  depressed  by  means 
of  the  nut  h  until  one  of  the  pivots  whose  irregularity  is  to 
?5e  measured  passes  freely  under  the  V's  aa.  Then  the  V's  hav- 
ing been  properly  adjusted  upon  the  pivot,  h  is  loosened  and 
the  flat  edge  of  the  aperture  in  /is  pressed  against  the  under 
side  of  the  pivot  by  the  spring  i.  The  elevation  of  the  rod 
g  above  the  glass  plate  is  then  measured  by  means  of  the 
spherometer.  This  consists  of  the  micrometer-screw  shown 
in  the  figure,  which  is  supported  by  the  small  tripod  s,  the 
legs  of  which  rest  on  the  glass  plate.  By  means  of  this  screw 
small  differences  in  the  elevation  of  the  rod  g,  and  conse- 
quently of  the  size  of  the  pivots,  may  be  readily  measured. 

Let  2v  =  the  angle  of  the  V's  aa\ 

n  =  the  difference  between  the  readings  of  the  screw 

on  the  two  pivots; 
R  =  the   linear   distance    between    two   consecutive 

threads  of  the  screw; 
L  =  the  distance  between  the  V's  of  the  transit  instru- 

ment; 
/  =  the  inequality  of  the  pivots  expressed  in  seconds 

of  time; 

r  —  the  radius  of  the  pivot  to  be  measured; 
C  =  the  distance  from  the  upper  surface  of  the  glass 

plate  to  the  angle  of  the  V's. 

Then  the  vertical  distance  from  the  upper  surface  of  the 
glass  plate  to  the  flat  surface  of  the  aperture  in  /will  be 


(298) 


sin  v  sin  v 

Similarly  for  the  other  pivot 

1  +  sin  v\  ,       . 

J-v—  —  .     .    .    (299) 
sin  v      i 


3O2  PRACTICAL   ASTRONOMY.  §  1  82. 

The  difference  ,s      (r  -  OL+£.  ......    (30o) 


This  is  evidently  the  difference  in  the  elevation  of  the  end 
of  the  rod  g  when  the  second  pivot  is  substituted  for  the 
first;  that  is,  the  difference  between  the  two  micrometer 
readings.  Therefore 


sin  v 
nR  sin  v 


on*       <30I> 


Then  from  c,  Art.  179. 

nR 


(302) 


This  instrument  is  especially  to  be  recommended  in  ex- 
amining the  pivots  for  irregularities,  as  by  measuring 
different  diameters  of  the  pivot  the  exact  form  may  be 
determined.  If  irregularities  exist  they  may  be  detected 
by  the  level,  but  it  will  not  show  which  pivot  is  irregular. 


The  Collimation  Constant,  c. 

182.  A  transit  instrument  of  the  better  class  is  provided 
with  a  micrometer,*  the  movable  thread  of  which  is  parallel 
to  the  threads  of  the  reticule  and  so  nearly  in  the  same  plane 
that  both  are  in  the  focus  of  the  eye-piece  at  the  same  time. 

*  For  description  of  micrometer  see  Art.  97. 


§  184.  THE   COLLIMATION  CONSTANT.  303 

With  this  arrangement  the  error  of  collimation  may  be 
measured  directly  as  follows : 

By  means  of  a  distant  terrestrial  object.  The  position  being 
clamp  west — suppose — direct  the  telescope  to  a  distant  ter- 
restrial point,  and  by  means  of  the  micrometer  measure  the 
distance  of  its  image  as  seen  in  the  field  from  the  middle 
thread,  then  reverse  the  instrument  and  measure  the  distance 
again.  If  the  object  appears  on  the  same  side  of  the  thread 
in  both  positions,  the  error  of  collimation  will  be  half  the 
difference  of  the  measured  distances ;  if  on  opposite  sides, 
half  their  sum. 

In  determining  c  in  this  way  care  must  be  taken  not  to 
mistake  its  algebraic  sign.  This  sign  may  be  determined 
practically  by  remembering  from  which  side  of  the  field  a 
star  at  upper  culmination  appears  to  enter.  If  then  for 
clamp  west  the  thread  appears  nearer  that  side  of  the  field 
than  for  clamp  east,  c  will  be  plus  for  clamp  west,  and  minus 
for  clamp  east. 

183.  By  the  collimating  telescope*     The  thread  or  cross- 
threads  of  a  collimating  telescope  may  be  used  in  the  same  way 
as  a  distant  terrestrial  object  for  measuring  the  collimation 
constant,  and  with  the  advantage  that  there  will  be  no  appre- 
ciable atmospheric  disturbance,  the  mark  being  only  a  few 
feet  distant.    With  two  collimating  telescopes,  one  north  and 
one  south  of  the  instrument,  the  error  may  be  determined 
without  reversing  the  instrument.     As  this  method  is  only 
of  practical  value  with  the  large  instruments  of  an  observa- 
tory, it  will  not  be  explained  further  here. 

184.  By    the    mercury    collimator*      If    the    telescope    is 
directed  vertically  downwards,  the  middle  thread  may  be 
seen  directly,  together  with   its   image  reflected   from  the 
mercury.     If  the  axis  is  horizontal  the  constant  c  will  be  one 

*  Art.  168. 


304  PRACTICAL  ASTRONOMY.  §  185. 

half  the  distance  between  the  direct  and  reflected  images, 
which  may  be  measured  as  before. 
If  the  axis  is  not  horizontal, 

Let  b  -=  the  elevation  of  the  west  end  ; 

M  —  the  micrometer  distance  of  the  thread  from  its 
image,  positive  when  the  thread  itself  is  on  the 
side  from  which  a  star  at  upper  culmination 
appears  to  enter. 

Then  \M  =  c  —  b\ 

........     (303) 


By  reversing  the  instrument  and  again  measuring  the 
distance  of  the  thread  from  the  reflected  image  we  can 
determine  both  b  and  c,  or  if  c  has  been  determined  by  the 
collimating  telescopes  we  can  determine  b  without  reversing 
the  instrument. 

185.  By  a  close  circumpolar  star.  With  the  portable  in- 
strument it  will  be  found  more  convenient  to  determine  the 
collimation  constant  by  observation  of  a  star  in  both  posi- 
tions of  the  axis,  as  follows  : 

Observe  the  transit  of  a  slow-moving  star  over  one  or 
more  threads  —  including,  the  middle  thread  or  not  —  then  re- 
verse the  instrument  and  observe  the  transit  over  the  same 
threads,  now  on  the  other  side  of  the  field.  With  one  of  the 
four  circumpolar  stars  of  the  Nautical  Almanac  there  will  be 
plenty  of  time  to  reverse  the  instrument  during  the  interval 
over  two  consecutive  threads.  It  is  advisable  to  read  the 
level  for  each  thread. 

The  times  observed  are  then  to  be  reduced  to  the  times 
over  the  middle  thread  (or  the  mean  thread,  as  the  case  may 
be)  by  means  of  the  equatorial  intervals,  which  must  be  well 
determined. 


§  1  86.  THE  AZIMUTH  CONSTANT.  305 

Let  T  —  the  clock  time  over  the  middle  (or  mean) 

thread  for  clamp  west  ; 
T  =  the  clock  time  over  the  middle  (or  mean) 

thread  for  clamp  east; 

b  and  b1  —  the  level  constants  in  the  two  positions  ; 
A  T  and  AT  —  the  clock  corrections  at  times  T  and  T  ; 
A  T0  =  the  clock  correction  at  time  T0  ; 
—  hourly  rate  of  clock. 


Then  AT  =  AT.  +  $T(T  -  T9); 

AT  =  ATQ  +  dT(T  --  T0). 


Then  applying  Mayer's  formula,  (284)  and  (285), 


Cl.W.  a  =  T  +  4T0+  ST(T-T0)  +  asin  (<p  -  tf)sec  <H 
+  b  cos  (<p—  #)  sec  #  -f-  c  sec  tf  —  S.o2i  cos  cp  sec  d  ;  I 

Ci.  E.  a  =  r+  4  ro+  8T(T—  T0)  +  a  sin  (<p  -  6}  sec  d  f 
+  V  cos  (9?  —  tf)  sec  (^  —  c  sec  #  —  S.O2  1  cos  cp  sec  d.  J  ' 

Subtracting  the  first  of  these  from  the  second,  we  readily  find 


c  =  KT  -  T)  cos  tf  +  \(T  -  T)  dTcos  tf 

+  K^  -  ^)  cos  (<p  -  d).    (305) 

This  formula  is  applicable  to  lower  culmination  by  chang- 
ing d  into  1  80°  —  8  as  usual.  In  most  cases  the  term  in  6T 
will  be  inappreciable. 


The  Azimuth  Constant,  a. 

186.  This  can  only  be  determined  by  observation  of  stars. 
Let  two  stars  be  observed  which  differ  as  widely  as  possible 
in  declination. 


306  PRACTICAL  ASTRONOMY.  §  1 86. 

Let  Tand  T'  be  the  times  of  observation  reduced  to  the 

middle  (or  mean)  thread; 
d  and  d',  the  declinations  of  the  stars; 
a  and  «',  their  right  ascensions. 

Then  equations  (304)  will  apply  to  these  stars,  except  that  in 
the  second  we  shall  have  a'  and  8'  in  place  of  a  and  #,  and 
the  sign  of  c  is  not  changed. 
Let  us  write 


/  =  T  -f-  8T(T  —  T0)  -f-  b  cos  (<p  —  8 )  sec  &  -{-  c  sec  # 

—  S.O2 1  cos  (p  sec  d  ; 
t'  =  T  +  6T(Tf-  T0)  +  b'  cos  (cp  -  cT)  sec  d'+  c  sec  V 

—  S.o2i  cos  cp  sec  (T. 

That  is,  we  place  t  and  /'  equal  to  the  sum  of  the  known 
quantities  in  the  second  members  of  the  equations.  Equa- 
tions (304)  then  become 


«=/-}-  AT.  +  a  sin  (9?  —  tf )  sec 
«'  =  t'  +  4T0  +  a  sin  (<p  —  d7)  sec 


From  which 


-    -    -  ,  ,, 

~  sin  (<p  -  6')  sec  $'  -  sin  (^  -  6)  sec  tf ' 
which  reduces  to 

(a'  -g)-  (f  -  t) 
~  cos^tantf-tand')" 

The  greater  the  denominator  of  this  fraction  the  smaller 
will  be  the  effect  upon  a  of  errors  of  observation.  If  two 
circumpolar  stars  are  observed,  one  at  upper  and  one  at 
lower  culmination,  the  denominator  of  (307)  becomes 

cos  cp  [tan  d  —  tan  (180°  —  #')]  =  cos  cp  (tan  8  +  tan  $'). 


§  l8/.  TO  DETERMINE  n  DIRECTLY.  3O/ 

This  combination  is  therefore  most  favorable  for  the  pur- 
pose.  If  the  rate  of  the  clock  and  the  stability  of  the  instru- 
ment can  be  relied  on  for  twelve  hours,  the  same  star  may 
be  observed  both  at  upper  and  lower  culmination.  This  will 
not  be  practicable,  however,  with  a  portable  instrument.  If 
two  stars  are  observed  at  upper  culmination,  one  should  be 
near  the  pole  and  the  other  near  the  equator. 

If  m  and  n  are  required,  they  may  now  be  computed  by 
(276),  or  we  may  proceed  as  follows. 


To  Determine  n  Directly. 

187.  Using  the  same  notation  as  in  the  determination  of  a, 
and  applying  Bessel's  formula,  (282), 

a  =  T  +  AT,  +  8T(T  —  T0)  +  m  +  n  tan  d  +  c  sec  8 

—  s.O2i  cos  cp  sec  #, 
af  =  Tf  +  AT,  +  6T(Tf  -  TQ)  +  m  +  n  tan  &  +  c  sec  <*' 

—  8.o2i  cos  cp  sec  <?', 

placing  the  known  terms  of  the  second  members  equal  to 
/  and  /'  respectively,  viz., 

t  —  T  +  <5T(T  --  ro)  +  c  sec  tf  —  8.o2i  cos  cp  sec  $, 
*t'  =  T  +  $T(T  —  ro)  +  c  sec  <?'  —  S.o2i  cos  cp  sec  <?', 

the  above  equations  become 

a  =  t  +  AT, 
a'  =  t' 


From  these  we  derive 

'  _     \  _'  _  /) 

-      •     •     •     •     (308) 


308  PRACTICAL  ASTRONOMY.  §  187 

Then  m  is  given  by  the  second  of  (277),  viz., 

m  —  b  sec  g>  —  n  tan  cp (3°9) 

The  conditions  favorable  for  an  accurate  determination  of 
n  are  evidently  the  same  as  in  the  case  of  a. 


Recapitulation    of  Formulas  for    Transit    Instrument   in   the 

Meridian. 


Equatorial  intervals,        i  =  I  cos  d  V  cos  /; 

*  =  /  cos  d. 

Reduction  to  middle       /  =  i  sec  d  t'sec  /; 
(or  mean)  thread,         /  =  i  sec  d. 

d 
Level  constant,  b  =  -  [W  —  E\. 


Collimation  constant,     c  =  $(T'  —  T)  cos 


'  —  T)8Tcos  S 
'-  J)  cos  (?>-*.) 


Azimuth  constant, 
Clock  correction, 


cos  <p(tan  ^  —  tan  d')' 

sin(^)  —  d)         cos(<p—  < 
cos  d  cos  6" 


,        c  s.O2i  cos  <p~| 

cos  6"  cos  d 


(XVII) 


For  reduction  by  Bessel's  formula  we  have  the  following  : 

(a'  —  a)  —  (tf  —  /) 


tan  6'  --  tan  » 
^  =  #  sec  cp  —  n  tan 
=  or  — 


— s.O2i  cos 


-  (XVIII) 


Transit  Observations. 

To  illustrate  the  application  of  (XVII)  let  us  reduce  the  following  observa- 
tions, made  at  the  Sayre  observatory,  1883,  October  16.  The  transit  is  a  small- 
sized  instrument  of  26  inches  focal  length,  aperture  2  inches ;  magnifying 


TRANSIT  OBSERVATIONS. 


309 


power  40  diameters.  The  reticule  contains  five  threads,  numbered  consecutively 
from  i  to  5  for  clamp  east.  As  will  be  seen,  the  level  was  generally  read  two 
or  more  times  in  each  position. 


1883,  October  16. 

Polaris.* 

Level. 

Level. 

8  =  88°  41'  23".8 

Clamp  West. 

Clamp  East. 

Clamp  west     V  oh  53™  34" 

E.                      W. 

E.                      W. 

IV  i      5     3i 

14.9               14.5 

13.0               16.7 

III  I    17     25 

13.0         16.6 

15-3               14-5 

Clamp  east    IV  i    29      3 

14.6         14.7 

13.0         16.8 

f             V  i    40     55 

13-0         16.5 

14.7         15.0 

14.7         14.7 

13.1         16.8 

Mean  clamp  W.  ih  17'"  238.4 

12.9         16.6 

14.8         14.9 

clamp  E.  i    17      7  .2 

13.0         16.8 

a  =  I    17     28  .83 

15-2       '14.3 

Mean  =  13.912     15.588 

13-983     15-783 

t/3  =  +  .838 

ft'  =  +  .900 

0  Arietis. 

V  Andromedae. 

S  =  20°  14'.  5 

d  =  41°  46'.  I 

I                45V 

I                 9'.  3 

II                  2.5 

II               31-2 

III                19.8 

III               52.9 

IV              37  -I 

IV              14  .8 

V  ih  48™  548.  5 

V  jh  57m  3?. 

T=  i   48     19  .78 

T  -  i    56     53  -04 

a  =  i   48     15  .35 

a  —  i    56    48  .81 

a  Arietis. 

£'  Ceti. 

d  =  22°  54'.8 

3  =  S°  i8'.2 

Level. 

L               S°.9 
II              26.2 

I              238.9 
II              40  .9 

E.                    W. 
14-7              15-3 

HI              43  -9 

HI              56.9 

12.5              17.9 

IV                i  .9 

IV              13  .4 

14.7              15-7 

V  2h     Im  I98.2 

V  2h    7m  29".  9 

12.7        17.8 

T  =  2     o    44  .02 

T  =  2     6     57  .00 

13.65     16.675 

a  =  2     o    39.54 

a  =  2     6     52  .35 

ft  =  +  1.512 

Instrument  reversed  for  the  purpose  of  determining  the  value  of  c. 


PRACTICAL  ASTRONOMY. 


y  Trianguli. 
I                528 
II                 II 
HI              31  . 
IV              50. 
V2hnm    9". 

•9 
i 

8 

5  Ursae  Minoris,  s.p. 
d  =  103°  47'  8" 
V 
IV              38°.4 
HI  2h  27m  47s.  3 

n          54.9 

I        3.5 

T  = 
a  = 

6 
1 
II 
III 
IV 

2     10 
2     10 

S  Ceti. 

=  —  oc 

3i  .14 
26.83 

io'6 

5s-  3 

21  .8. 

37-9 
54 

T=      2 

a  =  14 

8 
I 

III 

IV 

27      46  .85 
27      40.14 

y  Ceti. 

=  2°  44'.  8 

6'.9 
23 
39  -4 

Level. 

E.                     W 

12.6         17. 

15.0      15. 

12.6         17. 

9 

7 
9 

V 

2h  34'° 

I0«. 

9 

V 

2" 

37M 

558-9 

15 

.1 

15- 

8 

rr-  

2    33 

37 

.98 

T  = 

2 

37 

23  .12 

13 

.825 

16. 

825 

a  = 

2    33 

33 

•35 

a.  = 

2 

37 

18  .54 

ft 

—       1 

1.500 

<r8  Arietis. 

47  Cephei. 

S=i4° 

35' 

•9 

o  —  78°  57'  *8"                     Level. 

I 

37s. 

,2 

I 

2" 

48m 

i8 

E. 

w 

II 

54  • 

2 

II 

49 

27.8 

15. 

o 

15- 

4 

III 

10  , 

9 

III 

50 

51-5    * 

13. 

,6 

17- 

3 

IV 

27 

.8 

IV 

52 

18 

15. 

o 

15- 

3 

V 

2h  45m 

44s 

•9 

V 

2" 

53m 

428 

13 

7 

17- 

o 

T  = 

2   45 

ii  . 

00 

rr*   

2 

50 

52  .06 

14. 

325 

16. 

25 

a  = 

2   45 

6 

•57 

.a  = 

2 

50 

50  .41 

ft' 

=  + 

52 

The  values  of  the  apparent  right  ascensions  and  declinations  are  taken  from 
the  American  Ephemeris,  and  are  written  down  in  connection  with  the  observed 
transit  of  each  star,  a  must  be  taken  from  the  ephemeris  with  extreme  accu- 
racy, but  generally  6"  will  be  sufficiently  accurate  if  given  to  the  nearest  minute 
of  arc. 

Let  us  first  compute  the  values  of  the  equatorial  intervals  of  the  threads  by 
the  first  of  formulae  (XVII),  taking  for  this  purpose  the  observations  on  47  Cephei. 
The  numbers  in  the  first  column  of  the  following  table  are  obtained  by  subtract- 


REDUCTION  OF   TRANSIT  OBSERVATIONS. 


ing  the  observed  time  of  transit  over  each  thread  from  the  mean  of  the  times 
over  all  the  threads.  The  quantities  in  the  following  columns  will  require  no 
further  explanation. 

cos  d  =  9.28235. 


/. 

log/. 

log  %  cos  /.* 

log  /. 

f*. 

-j-  I7is.o6 
+    84.26 
+         -56 

2.23315 

1.92562 

9.74819 

9.99999 
9.99999 

1.51549 

1.20796 
9.03054 

+  32s.  77 
+  16.14 
+        -ii 

-    85  .94 

1.93420 

9.99999 

1.21654 

—  16  .46 

—  169  .94 

2.23029 

9.99999 

1.51263 

-  32  .56 

From  a  considerable  number  of  transits  the  following  values  of  the  equatorial 
intervals  were  finally  obtained: 


log  =  1.5.1359 

I.2I02I 
8.90309 
I.2I370» 


Clamp  east  ii  +  32s.  628 
ii  -j-  16  .226 

*8   -j-  .080 

it  -  16  .357 
*.  -  32  -588 

We  can  now  use  these  values  for  reducing  the  incomplete  transits  of  Polaris, 
5  Ursce  Minoris,  and  y  Ceti. 

In  cases  where  the  transit  is  observed  over  the  five  threads  the  arithmetical 
mean  is  taken. 

Let  us  compute  the  reduction  of  Polaris  in  full. 

cos  S  =  8.35913; 
log  sec  d  =  1.64087. 


log  z.      » 

log  ^sec.  /.* 

log/. 

/. 

Time  reduced  to 
Mean  Thread. 

Clamp  west. 

I.5I305 

.00078 

3.I547I 

4-  23m  470.9 

Ih  I7m  2I8.9 

1.21370 

.00020 

2.85477 

+  ii    55  -8 

17    26  .8 

S.gosogn 

.54396H 

-            3-5 

17     21  .5 

Clamp  east. 

I.2I370n 

.00020 

2.85477 

—  ii    55  -8 

I     17       7  .2 

I.5I305n 

.00078 

3.I547I 

—  23    47  .9 

i    17      7.1 

Clamp  west,  mean  ih  17™  23". 4; 
Clamp  east,  mean    1177  -2. 


*  See  table,  Art.  174. 


312 


PRACTICAL  ASTRONOMY. 


The  vame  of  /  used  in  taking  Vsec  I  frAn  the  table  is  obtained  by  subtract- 
ing the  times  of  transit  over  threads  Fand  //^respectively  from  the  time  over 
the  middle  thread.  Thus  we  have  from  the  observation 


Iv  -  23 


=  nm  54s. 


The  quantity  marked  ft  or  ft'  ,  in  connection  with  the  observations,  is  the 
inclination  of  the  axis  in  terms  of  one  division  of  the  level,  uncorrected  for  in* 
equality  of  pivots. 

From  the  first  level-reading  we  have 


Corrected, 
Therefore 


*/  = 
fi  = 
b  = 


-062. 
-900; 

-.157. 


=  ".174. 


The  value  of  b  used  for  those  stars  in  connection  with  which  the  level  is  not 
directly  read  is  obtained  by  interpolating  between  the  observed  values.  Thus 
we  have — 


STAR. 

f 

/3  corrected 
for/. 

b. 

Polaris,  clamp  west.  .  .  . 
Polaris,  clamp  east  
ft  Arietis 

+    .838 
+    .900 

.900 
.838 

+  M57 
.146 
167 

v  Andromedae      .        . 

188    ' 

a  Arietis  

.200 

€'  Ceti 

27O 

y  Trianguli 

+  1   ^12 

I  d^O 

2C2 

5  Ursae  Minoris,  s.p  
S  Ceti 

.252 

*     2m 

y  Ceti  

+  i  <;oo 

I  4.^8 

2CQ 

&  Arietis  

.204 

47  Ceohei 

+       062 

QOO 

IC7 

For  computing  the  error  of  collimation  c  we  have,  from  the  observed  transits 
of  Polaris, 


Clamp 

east 

T' 

—   jh  I7,r 

,      7s 

.2 

V  = 

•.146 

(p  =         40° 

36' 

24" 

Clamp 

west 

T 

=   I      17 

23 

•4 

b  = 

4-  -157 

d  =       88 

4i 

24 

T'  - 

T 

=   — 

16 

.2 

b'  -b  = 

—     .Oil 

tp  -  d  =  -  48 

5 

° 

*  Example,  Art.  179. 


§  187.         REDUCTION  OF   TRANSIT  OBSERVATIONS.  313 

log  K7"  -  T)  =  0.90849*  log  \(b'  -  b)  =  7-74036* 

cos  #  —  8.35913  cos  (<p  —  d)  =  9.82481 

sum  =  9.26762*  7-565i7* 

Nat.  No.        —  .1852  Nat.  No.        —  .0037 

Therefore  c  =  T  -.1889  clamp  j  ^  I . 

In  applying  the  formula  of  (XVII),  the  term  i(T'  —  T)  8T  cos  8  has  been 
disregarded,  as  in  this  case  it  is  inappreciable.  It  is  convenient  to  combine  the 
correction  for  diurnal  aberration  with  c. 

Thus,  if  we  write  c   —  c  —  s.O2i  cos  (p, 

we  have  in  this  case  c'  —  -}-  8. 173  clamp  east, 

c  •=•  —  ".205  clamp  west. 

The  last  but  one  of  (XVII)  will  now  give  us  the  azimuth  constant  a. 

We  have  seen  that  the  best  result  is  to  be  expected  when  we  use  the  observed 
transits  of  two  circumpolar  stars,  one  at  upper  and  the  other  at  lower  culmina- 
tion. We  therefore  determine  this  constant  from  5  Ursce  Minoris  and  47  Cephei. 

Referring  to  the  derivation  of  the  formula  for  a  (Art.  186),  we  have  for  /  and  t' 

t  =  T  +  b  cos  (cp  —  d)  sec  8  -f-  c  sec  8; 
t'  —  T'  -f-  b'  cos  (cp  —  d')  sec  d'  +  f'  sec  d'; 

the  term  in  8T-— the  rate — being  inappreciable. 
The  computation  is  then  as  follows: 

5  URS^E  MINORIS,  s.p. 

d  —       103°  47'    8"         log  sec  ==  0.62290*  =  log  C 
q>  =         40   36   24 
q>  —  d  =  —    63    10  44  log  cos  r=  9.65438 

Sum  =    .27728*  =  log  B 

b  =       O8.252  log  b  =  9.40140 

f'  =  +     -1'73  log  f'  =  9.23805 

Bb  =  —     .477  log  Bb  =  9.67868* 

Cc  =  -      .726  log  Cc'  -  9.86095* 

T  =  2h  27m  46s. 85 

Bb+  Cc'  =  —  i  .20 

/  =  2   27    45  .65 

47  CEPHEI. 

#'  =        78°  57'  18"  log  sec  =  0.71765  =  log  C 

(p  =        40   36   24 
<p  —  d'  —  —  38   20  54  log  cos  =  9.89446 

Sum  =    .61211  =  log  B 


314  PRACTICAL  ASTRONOMY.  §  l8/. 

b  =       o8. 157  log  b  —  9.19590 

c'  =  +  '*73  loS  *'  =  9-23805 

Bb  —  +  .643  ^  log  ^  =  9.80801 

Gr'  =  +  .903  log  Cc  -  9.95570 

T'  =  2h  som  52*  06  a'  =  2h  5om  5O8.4i 

Bb -\- Cc  —  +  i  .55  a=2   27    40.14 

/'  =  2    50     53 .61  a'  —  a  =       23     10  .27 

Nat  tan  8'  = -f-  5.1231  /'  —  t  —       23       7.96 

Nat  tan  8  =  —  4-0758          (a!  —  a)  —  (('  —  t)  =         -f-    2  .31 

tan  5  —  tan  5'  =  —  9.1989  log  =  0.96373;* 

cos  (f)  =  9.88036 
log  denominator  =  0.84409;* 
log  [(a'  -  a)  -  (/'  -  /)]  =    .36361 
a  =  —  8.33i  log  a  =  9.51952^ 

We  may  now  compute  the  clock  correction  A  T  from  the  last  of  formulae 
(XVII),  using  for  this  purpose  the  observed  transits  of  the  zenith  and.  equatorial 
stars.  We  require  first  the  values  of  the  coefficients. 

=  sin  (y  -  S)  B  =  «»(«>-«i.         and         c=       ' 


cos  d  cos  d  cos  S' 

If  the  instrument  is  to  be  much  used  at  any  one  place,  as  in  an  observatory 
for  determining  the  local  time,  it  will  be  very  convenient  to  tabulate  these1 
quantities  with  the  argument  d.  On  pages  220-227  °f  tne  U.  S.  Coast  Survey 
Report  for  1880,  Schott  gives  tables  of  these  factors  to  two  decimal  places,  with 
the  double  arguments  d  and  z  —  <p  —  S,  by  means  of  which  the  factors  may  be 
found  for  any  latitude  and  declination  within  the  limits  of  the  table.  If  such 
tables  are  not  at  hand,  a  computation  with  four-place  logarithms  will  give  the 
necessary  degree  of  accuracy ,t  The  work  may  be  arranged  as  follows: 

Star  ft  Arietis.                                                        Y  Andromedae. 

S  =  20°i4'.5  sin(<p— d)  =  9.5416  d=    4i°46'.i  sin  (<p—  d)  =  8.3o7» 

(^  =  4036.4  cos  d  =  9  9723  <p  =    4036.4  cos  d  =  9.8726 

cp— d  =  20  21  .9  cos  (<p—  d)  =  9.9720     (p— d  =  —  i    9.7  cos(<p— <5)  =  9.9999 

A—+    .371  log  A=  9.5693  A  =  —    .027  log  ,4  =8.434* 

B  =  +    .999  log  .5  =  9.9997  ^  =  +  1.341  log  B=    .1273 

C=  +  1.066  logC=    .0277  C  =  +  1.341  logC=    .1274 


§  187.         REDUCTION  OF   TRANSIT  OBSERVATIONS.  315 

The  determination  of  AT'\s  then  as  follows: 


STAR. 

A 

5 

C 

/«« 

Bb 

Cc' 

r 

a 

AT 

V 

(3  Arietis  

+  •37 
-.03 
+  •33 

+  •54 

±3 

+  .61 
+  •45 

1.  00 

i-34 
1.03 
•85 
1.19 
.76 
•79 
•93 

.07 

:2J 

.or 
.20 

.00 

.00 
•03 

—  .12 
+  .01 
—  .  II 
-.18 
—  .05 
—  .22 
—  .20 
-•15 

+  •17 
•25 
.22 
.20 

•3° 

.19 

.20 

.19 

+  .18 
•23 
.19 
•17 

.21 

•J7 
.17 

.18 

jh  4gm  I98>78 

i    56    53  -°4 
2      o     44  .02 
2      6     57  .00 
2    10    31  .14 
2    33     37  -98 
2    37     23  .12 
2    45     ii  .00 

ih  48"  I58.35 
56     48.81 
0     39  -54 
6     52  -35 
10    26  .83 
33     33  -35 
37     J8  .54 
45       6  .57 

-  4.66  -  8 

~  4'^T  2 
4.78+  4 

4.84+10 
4-77  +  3 
4-77+  3 
4-75+  i 
4-65  —  9 

y  Andromedae  

£'  Ceti  

v  Trianguli  
6  Ceti 

y  Ceti      

<ra  Arietis 

Mean  A!T  =  —  4*-744  ±  .022 

The  column  headed  v  contains  the  residuals  from  which  the  probable  error 
is  found  by  formula  (27)  or  (28). 


Application  of  Formula  (XVIII). 


These  formulae  will  not  often  be  used  for  reducing  observations  made  with 
an  instrument  of  this  class,  but  for  illustration  we  may  apply  them  to  the  above 
observations. 

Computation  of  n.  We  use  the  transits  of  5  Ursce  Minoris  and  47  Cepkei. 

t  =  T  -f  c'  sec  d         =  2h  27m  46*. 85  —  ".73         d  =  103°  47'    8" 
t'  -  T'  +  c'  sec  d'        =250    52.06+  .90        8'=    78    5718 


a'  =  2h  so01  so8. 41 

a.  =  2   27    40  . 14 

a'  —  a  —      23    10.27 

t'  -  t  =      23      7  .96 

(/'  _  4  =          +2.31 


tan  5'  =       5.1231 
tan  d  =  —  4.0758 

tan  d—  tan  d'  =  -f  9.1989 


For  ft  Arietis  b  =  -f  .167. 
Then  we  have, 


Therefore  w  =  -J-  ".373. 
Therefore  m  =  £  sec 


»  tan  <    =  —  8. 100. 


Arietis,         7*  =  ih  48™  I98.78 

m  —  .10 

n  tan  S  +  •  14 

/seed  +.18 

a       i  48    15.35 


AT-  -4".65. 


3*6  PRACTICAL  ASTRONOMY.  §  1 88. 


Personal  Equation. 

188.  When  the  results  of  transit  observations  made  by  dif- 
ferent observers  are  compared,  it  is  found  that  they  differ 
generally  by  small  but  nearly  constant  quantities.  One 
observer  perhaps  acquires  a  habit  of  noting  the  transit  too 
early  by  a  fraction  of  a  second,  while  another  will  note  it 
uniformly  too  late.  This  difference  is  called  the  personal 
equation.  It  is  customary  to  speak  of  the  relative  and  the 
absolute  personal  equation,  the  former  being  the  constant 
difference  between  the  right  ascensions,  or  clock  corrections 
deduced  from  observations  made  by  two  different  observers, 
and  the  latter  the  difference  between  the  absolute  value  of 
the  quantity  and  that  obtained  by  an  observer  who  notes  the 
time  uniformly  too  early  or  too  late.  When  results  obtained 
from  observations  of  two  different  observers  are  to  be  com- 
pared, as  in  the  determination  of  longitude,  the  personal 
equation  should  always  be  determined  and  the  necessary 
correction  applied. 

The  existence  of  a  large  personal  equation  is  not  an  indi- 
cation of  a  poor  observer,  but  perhaps  the  contrary.  Thus 
the  noted  observers  Bessel  and  Struve  found  that  in  1814 
their  relative  personal  equation  was  zero;  in  1821  it  was 
O8.8,  while  in  1823  it  amounted  to  an  entire  second:  thus  indi- 
cating the  gradual  formation  of  a  fixed  habit  of  observing  on 
the  part  of  both.  Also  in  1823  the  relative  personal  equa- 
tion between  Bessel  and  Argelander  was  is.2,  a  surprisingly 
large  quantity. 

The  personal  equation  also  depends  to  some  extent  on  the 
instruments  employed  and  the  method  of  observation.  It  is 
generally  much  smaller  when  the  chronograph  is  used  than 
when  the  eye  and  ear  method  is  employed.  Bessel  found 
that  when  he  used  a  chronometer  beating  half-seconds  he 


§  188.  PERSONAL  EQUATION.  317 

observed  transits  os-49  later  than  when  he  employed  a  clock 
beating  seconds. 

There  are  various  methods  of  determining  the  personal 
equation,  those  mosl  commonly  employed  being  the  follow- 
ing: 

First  Method.  Let  one  observer  note  the  transit  of  the  star 
over  the  first  two  or  three  threads, 'and  the  other  observer  its 
transit  over  the  remaining  threads.  The  observed  times  are 
reduced  to  the  middle  (or  mean)  thread  by  means  of  the 
equatorial  intervals,  and  the  difference  of  the  reduced  times 
will  be  the  relative  personal  equation. 

A  considerable  number  of  stars  should  be  observed  in  this 
way,  each  observer  leading  alternately.  Among  the  various 
methods  used,  this  is  considered  one  of  the  most  reliable. 

Second  Method.  The  two  observers  may  each  use  a  different 
instrument  and  determine  the  clock  correction  separately,  ob- 
serving the  same  list  of  stars.  When  the  instruments  which 
the  observers  are  accustomed  to  use  differ  considerably  in  the 
arrangement  of  the  threads  or  in  other  respects,  this  method 
may  be  superior  to  the  former,  as  each  observer  may  use  his 
own  instrument  and  make  his  observations  deliberately  and 
in  his  usual  manner. 

Third  Method.  By  a  personal-equation  apparatus.  Various 
mechanical  devices  have  been  constructed  for  measuring 
both  the  relative  and  absolute  personal  equation.  Prof.  Hil- 
gard  describes  two  machines  of  this  kind  in  Appendix  17, 
Coast  Survey  Report  1874.  An  instrument  designed  by 
Prof.  Eastman  has  been  in  use  at  the  Naval  Observatory  for 
a  number  of  years,  for  a  description  and  drawing  of  which 
see  Appendix  I,  Washington  Observations,  1875.  These  all 
consist  of  a  mechanical  device  for  causing  an  artificial  star 
to  pass  across  a  field  of  view  arranged  to  appear  as  nearly 
as  may  be  like  that  of  the  transit  instrument.  The  observer 
notes  the  time  of  transit  across  the  threads  either  by  the 


3l8  PRACTICAL   ASTRONOMY.  §  189. 

chronographic  or  the  eye  and  ear  method,  while  the  machine 
by  an  electric  arrangement  records  the  time  automatically, 
constant  differences  between  the  actual  time  of  transit  and 
that  recorded  by  the  machine  being  eliminated  by  causing 
the  star  to  cross  the  field  in  both  directions.  The  difference 
between  the  automatic  record  and  that  of  the  observer  is  his 
absolute  personal  equation. 

Prof.  Eastman  gives  the  following  examples  of  the  relative 
personal  equation  deduced  on  the  same  night  by  this  instru- 
ment and  by  method  first: 

By  By  Ap- 

btars.  paratus. 

October       25,  1875,  Professor  Eastman — Assistant  Skinner.  ..os.25i  o9.227 

November    5,  1875,  Professor  Eastman— Assistant  Paul 174  .173 

December     6,  1876,  Professor  Eastman — Assistant  Paul 035  .052 

December  31,  1877,  Professor  Eastman — Assistant  Frisby 052  .044 

March         13,  1878,  Professor  Eastman — Assistant  Frisby 052  .054 

March         23,  1878,  Professor  Eastman — Assistant  Paul 107  .092 

This  close  agreement  between  the  results  obtained  by  two 
methods  so  entirely  different  must  be  regarded  as  exceed- 
ingly satisfactory. 

The  observer's  physical  and  mental  condition  is  sometimes 
found  to  exert  a  marked  influence  upon  his  personal  equa- 
tion. It  is  therefore  very  desirable  that  while  prosecuting 
observations  where  great  accuracy  is  essential  he  should  main- 
tain as  far  as  possible  his  ordinary  habits  of  mind  and  body. 

In  the  more  accurate  longitude  work  of  the  Coast  Survey 
the  effect  of  personal  equation  is  eliminated  by  the  observers 
exchanging  stations  when  the  work  is  about  half  finished. 

Probable  Error  and  Weight  of  Transit  Observations. 

189.  The  probable  error  of  an  observed  transit  consists 
practically  of  two  parts:  first,  the  probable  error  of  the 
observer  in  noting  the  time  of  the  stars  passing  the  threads, 
independent  of  his  personal  equation;  and  secondly,  the  vari- 


§  '.89.    PROBABLE  ERROR   OF  TRANSIT  OBSERVATIONS.      319 

ous  errors  which  together  form  what  is  known  as  the  culmi- 
nation error.  Among  these  latter  are  those  due  to  atmos 
pheric  displacement,  outstanding  instrumental  errors,  irreg- 
ularities of  the  clock  rate,  and  changes  in  the  personal  equa- 
tion. The  culmination  error  is  not  diminished  by  increasing 
the  number  of  threads  of  the  reticule. 

The  first  part  of  the  probable  error,  which  for  present 
purposes  we  may  call  the  personal  error,  may  be  determined 
by  comparing  together  the  individual  values  of  the  equa- 
torial intervals  deduced  from  a  large  number  of  observations, 
using  for  the  purpose  the  formula 


=  .6745 


m  being  the  whole  number  of  determinations. 

Let  e  =  the  probable  error  of  the  observed  time  of  an 
equatorial  star  over  one  thread. 

Then,  since  the  equatorial  interval  is  the  difference  of  two 
observed  quantities,  each  of  which  has  the  probable  error  et 
we  shall  have  (Eq.  29) 


from  which 


As  the  result  of  the  discussion  of  a  large  number  of  obser- 
vations made  with  the  different  instruments  of  the  Coast 
Survey,  Schott  gives,*  for  the  larger  instruments, 


+  (.036)'  tan2  tf;  .     .     .     .     (311) 


Coast  Survey  Report  for  1880,  p.  236. 


320  PRACTICAL  ASTRONOMY. 

and  for  the  smaller  instruments, 


e  =  V(.oSoy  +  (.063)'  tan2 


§190. 


(312) 


From  these  equations  the  probable  error  for  a  star  of  any 
declination  may  be  computed,  and  consequently  the  weight, 
by  (33).  The  following  table  is  from  the  Coast  Survey 
Report,  the  weight  of  an  equatorial  star  being  unity : 


2 

For  large  portable  transits. 

For  small  portable  transits. 

e 

* 

#: 

6 

P 

& 

0          / 
0 

s. 

±  0.06 

i 

i 

S. 
±  0.08 

i 

I 

* 

10 

.06 

i 

i 

.08 

0.98 

I 

20 

.06 

0.98 

i 

.08 

.92 

0.96 

3° 

.07 

.91 

0.95 

.09 

•83 

.91 

4° 

.07 

.82 

.90 

.  o 

.70 

•83 

45 

.07 

•76 

.87 

.    0 

.62 

•79 

5° 

.08 

.69 

•83 

.  I 

•53 

•73 

.08 

.61 

•78 

.    2 

•44 

.66 

60 

.09 

•51 

•71 

•  4 

•34 

•59 

65 

.10 

.40 

•63 

.  6 

.26 

•Si 

70 

.12 

.29 

•54 

.19 

.18 

.42 

75 

•  J5 

.18 

•43 

25 

.10 

•32 

80 

.21 

.09 

•30 

•37 

.05 

.22 

85 

.42 

.02 

•IS 

.72 

.01 

.11 

5  Ursae  Minoris... 

86    36 

0.61 

O.OII 

0.103 

i.i 

0.006 

0.075 

51  Cephei  
a  Ursae  Minoris.  .  . 

87     14 
88     39 

0.75 

0.007 
O.OO2 

0.084 
0.041 

2.7 

0.004 

O.OOI 

O.o6  1 
0.030 

A  Ursae  Minoris.  .  . 

88     56 

1.9 

0.001 

0.033 

3-4 

0.001 

0.024 

In  the  application  of  the  multiplier  Vp  it  generally  suffices 
to  employ  but  one  significant  figure. 

Relative  Weights  of  Incomplete  Transits. 

190.  Let  s  —  the  probable  error  of  the  transit  of  an  equa- 
torial star  over  a  single  thread  ; 
f,  =  the  probable  culmination  error ; 
r  =  the  probable  error  of  the  transit  observed 
over  n  threads,  both  sources  of  error  being 
considered 


p  IQO.  WEIGHTS  OF  INCOMPLETE    TRANSITS.  $21 

Then  r*=e?  +  ~  ........     (313) 

Schott  concludes,  from  the  examination  of  558  individual 
values  of  the  right  ascensions  of  36  stars  observed  at  the 
U.  S.  Naval  Observatory,  that  for  the  larger  instruments  of 
the  Coast  Survey  r  =  os.o5i,  and  for  the  smaller  instru- 
ments r  =  O8.o6o.  When  assigning  to  £  the  values  os.o63  and 
os.o8o  from  (311)  and  (312),  it  is  found  that  e,  =  ±  ".049  and 
±  S.os6  respectively.  Then  let 

N  =  the  whole  number  of  threads  ; 
p  =  the  weight  of  an  observation  over  n  threads  ; 
Unity  =  the  weight  of  an  observation  over  all  of  the  ^threads. 


Then,  (33),  p=  --  -  ......    (314) 


Substituting  the  above  values  for  s  and  €lt  we  have  — 


For  the  larger  instruments  p  —  —    —  ^;  .....    (315) 


,     2.0 

I+AT 

For  the  smaller  instruments  p  =  -        — (31 6) 


322 


PRACTICAL   ASTRONOMY. 


Let  N  —  25  in  (315)  and  9  in  (316)  respectively;  we  find 
the  following  values  of/  for  the  values  of  n  indicated. 


n 

P 

n 

P 

I 

.41 

13 

•95 

2 

•59 

14 

.96 

3 

.69 

15 

.96 

4 

•  76 

16 

•97 

5 

.81 

17 

•97 

6 

.84 

18 

.98 

7 

.87 

19 

•98 

8 

.89 

20 

.98 

9 

.90 

21 

•99 

10 

.92 

22 

•99 

ii 

•93 

23 

•99 

12 

•94 

24 

I.OO 

25 

1.  00 

n 

P 

i 

.41 

2 

.61 

3 

•73 

4 

.82 

5 

.87 

6 

.92 

7 

•95 

8 

.98 

9 

I.OO 

It  appears,  therefore,  that  the  gain  in  accuracy  obtained  by 
increasing  the  number  of  threads  soon  becomes  practically 
insignificant.  Bessel  thought  that  no  practical  advantage 
resulted  from  the  use  of  more  than  five  threads. 

Reduction  of  Transit  Observations  by  Least  Squares. 

191.  When  the  time  is  to  be  determined  by  a  series  of 
observations  with  the  portable  transit  instrument,  the  method 
of  least  squares  may  be  applied  with  advantage  in  case  the 
results  are  required  with  extreme  accuracy.  This  will  be 
the  case  particularly  where  the  time  is  required  for  longitude 
determination,  and  where  the  clock  correction,  the  azimuth 
and  collimation  constants,  and  sometimes  the  rate,  are  all  to 
be  determined  from  the  same  series  of  observations. 

An  observing  list  should  be  prepared  beforehand,  embrac- 
ing stars  adapted  to  the  determination  of  these  quantities. 
We  have  seen  that  stars  which  culminate  near  the  zenith  are 
best  adapted  to  the  determination  of  AT\  also  that  circum- 


§  191.         REDUCTION  OF  TRANSIT  OBSERVATIONS.  323 

polar  stars  observed  at  upper  and  lower  culmination  are 
best  for  the  determination  of  a.  One  half  the  stars  should 
be  observed  in  each  position  of  the  axis  for  the  purpose  of 
determining  c. 

It  is  a  very  good  arrangement  to  divide  the  stars  into 
groups  of  about  five  or  six  stars,  each  group  to  contain 
two  circumpolar  stars,  one  at  upper  and  one  at  lower 
culmination,  the  remaining  three  or  four  stars  being  near  the 
zenith  or  between  the  zenith  and  equator.  It  is  not  advis- 
able to  include  the  close  circumpolar  stars  in  such  a  group. 

The  instrument  having  been  carefully  adjusted,  the  observa- 
tions will  be  conducted  as  follows : 

ist.   Read  the  level. 

2d.    Observe  the  first  group  of  five  or  six  stars. 

3d.    Read  the  level. 

4th.  Reverse  the  instrument. 

5th.  Read  the  level. 

6th.,  Observe  the  second  group  of  five  or  six  stars. 

;th.  Read  the  level. 

This  may  be  regarded  as  a  complete  series,  as  it  contains 
everything  necessary  for  determining  all  of  the  unknown 
quantities.  If  considered  desirable,  a  third  and  fourth  group 
may  be  observed  in  the  same  manner.  If  there  is  time  be- 
tween the  stars  of  the  group,  more  leveLreadings  may  be 
taken ;  but  if  the  mounting  is  reasonably  firm,  the  level 
corrections  for  the  individual  stars  may  be  interpolated  from 
those  at  the  beginning  and  end. 

If  there  are  no  imperfect  transits,  a  knowledge  of  the 
equatorial  intervals  will  not  be  required  ;  otherwise  they  may 
be  determined  from  the  suitable  stars  of  the  series  just  ob- 
served. It  must  be  remembered  that  in  transporting  the 
instrument  from  one  station  to  another  the  relative  position 


324  PRACTICAL   AS7^RONOMY.  §  191. 

of  the  threads  is  liable  to  be  disturbed.     This  difficulty  is 
avoided  by  the  use  of  the  glass  reticule,  the  distances  of  the 
lines  of  which  may  be  determined  once  for  all. 
The  reduction  is  then  as  follows  : 

Let  A  =  sin  (<p  —  d)  sec  6  ; 
B  =  cos  (q>  —  6)  sec  d  ; 
C  =  sec  d  ; 

A  T0  =  the  clock  correction  at  time  T0  ; 
dT  =  the  hourly  rate  ; 
a  =  the  stars'  .apparent  right  ascension. 

We  can  always  infer  from  our  observations  a  value  of  A  T0 
which  will  be  very  near  the  true  one,  and  as  the  labor  of 
computation  will  be  diminished  by  making  the  numerical 
values  of  the  unknown  quantities  as  small  as  possible,  we  may 
assume  an  approximate  value  of  this  quantity,  and  determine 
a  correction  to  this  assumed  value. 

Let     5  =  the  assumed  value  of  the  clock  correction  ; 


Then  x  is  a  small  unknown  correction  to  5. 

Introducing  this  notation  into  Mayer's  formula,  it  becomes 


In  which  x,  6T,  a,  and  c  may  be  considered  unknown  quan- 
tities. 

Writing     /=  T  +  $  +  Bb  —  s.O2i  C  cos  <p  —  a, 

viz.,  the  sum  of  the  known  quantities,  we  have 

Aa+Cc+ST(T-  r.)  +  *  +  /  =  o.       .    (317) 


§  IQI.         REDUCTION  OF   TRANSIT  OBSERVATIONS.  325 

Every*observed  transit  furnishes  one  equation  of  this  form 
for  determining  the  four  unknown  quantities  a,  c,  $T,  and  x. 
Four  perfect  observations  would  be  sufficient.  As  a  much 
larger  number  will  be  taken,  the  most  probable  values  must 
be  determined  by  the  method  of  least  squares  (Art.  21). 

If  8T  is  known,  the  number  of  unknown  quantities  will  be 
reduced  to  three.  If  in  addition  c  has  been  determined  by 
some  other  method,  there  will  only  be  two. 

If  there  is  a  suspicion  that  the  azimuth  has  changed  dur- 
ing the  progress  of  the  observations,  an  additional  azimuth 
constant  may  be  introduced  as  another  unknown  quantity. 

The  reduction  will  be  facilitated  by  tabulating  the  factors 
A,  B,  and  C.  Such  a  table  has  been  published  by  the  U.  S. 
Coast  Survey,  in  which  A  and  B  are  given  with  the  double 
argument  d  and  z  —  (cp  —  #).  C  is  of  course  given  with  the 
argument  tf. 

When  many  observations  are  to  be  reduced  at  one  place, 
or  in  the  same  latitude,  a  special  table  is  more  conveniently 
computed  for  the  latitude  of  the  place.  The  only  argument 
will  then  be  S. 

It  will  be  convenient  to  make  the  computation  of  /  directly 
in  the  book  used  for  recording  the  transits.  The  means  of 
the  times  over  the  threads  being  taken,  this  will  be  T,  which 
is  written  below.  In  case  of  incomplete  transits,  the  time 
over  the  mean  thread  is  computed  as  already  illustrated,  a 
and  $  are  taken  from  the  Nautical  Almanac  and  written  in 
thesamebook.  The  small  corrections^.^  and  — S.o2i  cosp.C 
are  applied  directly  to  T.  Subtracting  a  from  the  algebraic 
sum,  we  have  /  —  5,  in  which  3  will  be  assumed  of  such  value 
as  to  make  /  small.  An  example  follows. 


PRACTICAL  ASTRONOMY. 


Reduction  of  Transit  Observations  made  at  the  Sayre  Observatory,  1883,  October  n. 
An  observing  list  was  first  prepared,  of  which  the  following  is  a  specimen  : 


STAR. 

Magnitude. 

a 

£ 

Setting. 

u.  Aquarii 

47 

2O*1   46m    2  18 

—       O°    2<>'   3 

I4O°       l'   7 

•y  Cygni          

4  O 

2O      52      4Q 

4O     4^    O 

80      5^    4 

o"2  Ursae  Majoris,  s.p.  .. 
r  Cvcrni 

5-0 

•j  o 

21         0         5 

21         7      57 

112      23.5 
2Q      44    8 

18     12  .9 

IOO      51    6 

r  Cvcrni 

A    O 

21      IO        7 

•77      -32    8 

Q-3               O       A 

ex.  Cephei  

2.7 

21      15      47 

62         54 

68     31  o 

£  Pegasi         .... 

2  ^ 

21      38       26 

9      2o    3 

121       l6    I 

7^2  Cygni  

4.  3 

21      42      28 

48      46    I 

81     50  ^ 

79  Draconis  

6  3 

21       51       25 

7-1       8  .0 

57      27    5 

ex.  Aquarii 

q  o 

21       5Q      46 

—       O      ^    ^ 

1^1       2Q    7 

32  Ursae  Majoris,  s.p... 

6.0 
4  7 

22         9      31 
22      19       l8 

114    18.5 
-j-    o    47  .0 

16       17.9 
I2Q      4Q  .4. 

The  two  groups  are  intended  to  be  observed  one  in  each  position  of  the  axis. 
The  right  ascension  and  declination  are  taken  from  the  mean  values  of  the 
Nautical  Almanac.  The  column  headed  ' '  Setting"  gives  the  setting  of  the  finding 
circle.  In  this  case  the  circle  reads  zero  when  the  telescope  is  directed  to  the 
north  point  of  the  horizon,  the  latitude  being  40°  36' 24";  the  circle  will  read 
130°  36'  24"  when  the  line  of  collimation  of  the  telescope  lies  in  the  equator. 
Therefore  the  setting  for  any  star  will  be  130°  36'. 4  —  d. 

Below  is  the  copy  of  the  recorded  transits  of  the  above  stars  as  observed  on 
the  night  of  October  n,  1883  : 


Level. 

E.  W. 

12.0  9.9 

9-2  I3-I 

12.0  9.9 

9-6  13.0 

10.70  11.475 


Clamp  East. 

/ 

i  Aquarii. 

I 

57- 

II 

13-9 

III 

30. 

IV 

46.7 

V 

20  47     3.1 

I 

II 

III 

IV 

V 


v  Cygni. 

14-4 
36.3 

57-5 
19. 
20  53  40.4 


T  =  20   46   30.14  -j-  .02 

a  =  20  46  24.07 

£  Cygni. 
I  28.9 


a2  Ursse  Majoris,  s.p. 

V  49.9 

IV  31.2  II  48. 

Ill  14.8  III  6.8 

II  IV  25.8 

I  21    I    40.  V         21    8  44.1 

T  —  21  o  14.62  —  .01         T  =  21  8  6.72 

a  =  9  o     7.86  a  —  21  8  0.69 


I 

II 

III 

IV 

V 


T  —  20  52  57.52  -|-.o6 
a  =  20  52  51.77 

T  Cygni. 


56 

16.1 

36.9 

21  10  57.6 


.05 


T  =  21  10  16.36  +  .06 
a  =  21  10  10.56 


§I9I 


REDUCTION  OF   TRANSIT  OBSERVATIONS. 


327 


I 

II 

III 

IV 

V 


Cephei. 


Level. 


21    17 


46.1 
21.7 
55-9 
30.9 
5-9 


T=  21  15  56.10  +  . ii 
a  =  21  15  50.57 


E. 

9.8 
13-7 

9-3 
12.9 

9-5 

12.8 


w. 
13.3 

9.9 
I4.I 

10.8 
14.1 

II. 2 


11.333      12.233 


Level. 


10.2 

13-5* 

I2.9 

II.  2 

104 

I3-I 

12.7 

11.4 

n-55 


12.30 


Clamp  West. 

e 

Pegasi. 

V 

3«I 

IV 

I9.2 

III 

36. 

II 

52.8 

I 

21  39 

rri   

21    38 

36.04  + 

a  = 

21    38 

30.00 

V 
IV 

Ill 
II 

I 


r*  Cygni; 

48.9 
13.3 

38.1 

2-7 

21  43  27.5 


.05  T  —  21  42  38. 10 +  .11 

a  =  21  42  31.96 


79  Draconis. 


Level. 


V 

44. 

IV 

38.7 

III 

21  51  35-8 

II 

31-5 

I 

27.8 

T  =  21  51  35.56  +.23 
a  =  21  51  29.26 


E. 

10.2 
12.6 

10.5 

12.8 


w. 

13.9 

11. 2 
13.7 

11. 3 


11.525      12.525 


V 
IV 
III 

II 

I 


a  Aquarii. 


23-9 
40. 
56.8 
I2.9 
o  29. 


T=  21  59  56.52-}-. 06 
a  =  21  59  50.21 


32  Ursse  Majoris,  s.p. 


I 

II 

III 

IV 

V 


22 


T=  22 

a  =  10 


19. 

59-5 

9  38.5 

18.5 

57-5 


9  38.60  —  .04 
9  32.66 


V 
IV 
III 

II 
I 


TT  Aquarii. 

55-5 
11.9 
28.2 
44.1 

22    20      0.3 


T  —  22    19    28.00 

a  —  22  19  21.93 


.  08 


Level. 


12.6 
10. 1 
12. 1 

10.6 


H-35 


II. 2 
14.0 

n.8 
13-6 


12.65 


The  small  quantities  added  to  T  above  include  the  corrections  for  level  and 
diurnal  aberration;  viz.,  Bb  —  8.o2i  C,  cos  <p.  b  is  computed  from  the  level- 
readings  as  already  explained,  the  value  of  one  division  of  the  level  being  '.174, 

(  F    ) 
and  the  correction  for  inequality  of  pivots  being  T  .062  Cl.  \  ^  K  expressed 

in  terms  of  one  division  of  the  level. 


328 


PRACTICAL  ASTRONOMY. 


We  now  take  from  the  tables  the  values  of  the  coefficients  A,  B,  and  C,  or,  if 
tables  of  these  quantities  are  not  at  hand,  we  compute  them  by  the  formulae. 
For  illustrating  the  application  of  the  proper  weights  to  the  equations  of  con. 
dition,  the  value  of  Vp  is  taken  from  the  table  of  Art.  189  for  the  smaller  instru- 
ments. All  these  quantities  are  conveniently  tabulated  as  follows: 


STAR. 

1 

A 

B 

C 

Level. 

b 

Clamp  East. 
ju.  Aquarii    

—     9°     24'.  9 

.78 

.65 

I.OI 

+•326 

4-8  057 

v  Cygni  
<r2  UrsaeMajoris,s.p. 
£  Cygni        .   . 

40°     43'.6 

112°       24'  .0 

29°     45'.  4 

.00 

2.49 

.22 

1.32 
-  .82 
E.I3 

1.32 
—  2.62 
X.I5 

•059 
.061 
061 

07°          OQ  '.4 

126 

i  26 

.UUj 

a  Cephei  

62°      6  .0 

—    .78 

1.99 

2.14 

+.088 

068 

Clamp  West. 
e  Pegasi  
7T2  Cygni  

9°     20'.  8 
48°     46'.  7 

•53 

—    .22 

.87 
1.50 

—   I.OI 

—  1.52 

+.437 

.076 
.087 

79  Draconis    

73°       9'-5 
—    o°     52'  8 

-1.86 
66 

2.91 

-  3-45 

+.562 

.098 

106 

32  UrsaeMajoris,s  p. 

114°     ig'.o 

2.^7 

—  .68 

~f~  2-43 

TT  Aquarii  

o°     47'  5 

.64 

•77 

—  i.oo 

+.712 

.124 

STAR. 

Bb 

Aberration. 

Sum. 

#, 

1-4 

• 

Clamp  East. 

(*.  Aquarii  

+  •.04 

—  .02 

+  .02 

I.OO 

—  6".o9 

—  .09 

v  Cygni            

.08 

—  .02 

+  .06 

.82 

-  5  .81 

+  .19 

o-2  UrsaeMajoris,s.p. 
£  Cygni  

—    -05 
+  -07 

+  .04 
—  .02 

—  .or 
+  .05 

.46 

-6.75 
-6.08 

i  a 

T  Cvgni          

.08 

—  .02 

+  .06 

•85 

-  5.86 

"4~  14 

a  Cephei        

•«4 

+  .ii 

•56 

+  •36 

Clamp  West. 

e  Pegasi  

.07 

—  .02 

+  -05 

I.OO 

—  6  .09 

—  .09 

*2Cygm.:  
79  Dradoms. 

•  13 
.29 

—  .02 
—  .06 

T  -23 

•74 

36 

=6:2553 

—•25 

a  Aquarii    
32  UrsseMajoris,s.p. 

.08 
—  .08 
.10 

—  .02 
+  -04 
—  .02 

+  .06 
—  .04 
T  -08 

I.OO 

•5° 

-6.37 
—  5-90 

—  -37 

-+    .10 

Assumed  d  =  —  6«. 


The  quantity  in  the  column  headed  /  —  5  is  obtained  by  adding  algebraically 
to  the  quantity  T  of  the  above  observations  the  sum  of  the  corrections,  viz., 
Bb  —  8.02i  C  cos  q>,  and  subtracting  from  the  result  a.  We  now  have  all  the 
quantities  entering  into  the  equations  of  condition,  each  of  which  has  the  form 


§191.         REDUCTION  OF  TRANSIT  OBSERVATIONS.  329 

The  rate  is  here  inappreciable,  and  the  term  dT(T—T0)  has  accordingly  been 
dropped. 

The  coefficient  f,  as  will  be  seen,  has  its  sign  changed  for  clamp  west. 
Our  twelve  equations,  written  out  in  full,  will  then  be  as  follows  : 

1.  .780  -+•  i.oic  +  i.  OOP  —  —  .09. 

2.  .000  +  i.o&c  +    .82.*:  =:  -f  .16. 

3.  1.150  —  i.2i*  +    .46-^  =  —  -35- 

4.  ,2oa  -j-  1.05*:  -|-    igi-r  =  —  .07. 

5.  .060  -|-  1.07*:  +    .85*  =  +  .12. 

6.  —.44^  +  1.  20*  +     .56*  =  +  .20. 

7-         -53#  ~~  i.  oic  -}-  i.  oar  =  —  .09. 

8.  —  .i6a  —  1.  12*  +    .74*=  —  .19. 

9.  —.6ja  —  1.24*  +    .36*  =  —  .19. 

10.  .660  —  i.oor  +  i-  oar  =  —  .37. 

11.  i.ida  -{-  i.2ic  -\-    .$ox  —  -f-  .05. 

12.  .640  —  i.  oo*  -{-  i-oox  =  —  .15. 

These  now  have  the  general  form  of  the  equations  of  condition  (36),  viz., 
<*ix  -f  erf  +  d\w  —  »i, 

there  being  in  this  case  the  three  unknown  quantities  a,  c,  and  x,  correspond- 
ing to  the  x,  z,  and  w  of  the  general  form.  The  term  corresponding  to  y  has 
disappeared  here,  as  we  have  assumed  the  rate  of  the  clock  to  be  inappreciable 
for  the  short  time  over  which  the  observations  extend. 

We  have  now  to  form  the  normal  equations  (see  Eq.  41).  In  order  that  no 
confusion  may  arise  from  the  difference  of  notation,  the  general  form  of  these 
equations  is  here  given  in  full,  viz.: 

[aa]a  -f-  [ac\c  -f-  |W];r  =  [aw]  ; 

Ma  +  [«>  +  M*  =  M  ; 

[ad]a  +  [cd]c 


We  shall  give  the  solution  of  these  equations  in  full  with  the  various  checks  on 
the  accuracy  of  the  computation,  as  an  illustration  of  the  method.  Practically, 
however,  this  part  of  the  work  will  generally  be  more  or  less  abridged  by  ex- 
perienced computers  when  the  number  of  unknown  quantities  does  not  exceed 
that  of  the  above  equations. 

We  shall  require,  besides  the  quantities  already  indicated,  the  sums  of  the 
coefficients  of  each  equation,  viz.: 


33°  PRACTICAL  ASTRONOMY.  §  IQI. 

Also,  we  compute  the  quantities 

[as],  [cs],  [ds],  [nn],  [ns]. 

The  computation  will  first  be  made  by  the  use  of  Crelle's  table. 

We  therefore  prepare  the  scheme  for  computation  given  below,  containing  TO) 
columns,  5  for  the  quantities  a,  c,  d,  —  n,  s,  etc. ,  which  we  rewrite  for  the  sake  of 
convenience,  and  14  for  the  squares  and  products. 


a. 

c. 

d. 

—  n. 

s. 

aa. 

ac. 

ad. 

—  an. 

as. 

cc. 

.78 

.01 

I.OC 

+  .09 

2.88 

.6084 

+  .7878 

+  .7800 

+  .0702 

+  2.2464 

.0201 

.00 

.08 

.82 

—  .  16 

J    JA 

—  .0000 

—  .0000 

.1664 

1.15 

—   .21 

+  .35 

•75 

1.3225 

—  I-39I5 

+  .5290 

+  .4025 

.8625 

.4641 

.20 

•°5 

•  91 

+  .07 

2.23 

.0400 

+  .2100 

+  .1820 

+  .0140 

.4460 

.1025 

.66 

.07 

•  8  = 

—  .12 

1.86 

.0036 

+  .0642 

+  .0510 

—  .0072 

.  1116 

.1449 

—  •44 

.20 

3 

—  .20 

I.  12 

.1936 

—  .5280 

-  .2464 

+  .0880 

-  .4928 

.4400 

•  53 

—   .01 

I.OC 

+  .09 

.61 

.2809 

-  -5353 

+  .53°° 

+  -0477 

•  3233 

.0201 

—  .16 

.12 

•  74 

+  .19 

—  -35 

.0256 

+  .1792 

-  .1184 

—.0304 

.0560 

•2544 

-.67 

—   .24 

•3" 

+  .19 

—  1.36 

.4489 

-f-  .8308 

—  .2412 

—.1273 

.9112 

•5376 

.66 

—   .00 

I.OC 

+  •37 

1.03 

•4356 

—  .6600 

+  .6600 

+  .2442 

+  .6798 

.OOOO 

1.16 

+   .21 

•  5C 

2.82 

I-3456 

+  1.4036 

+  .5800 

—.0580 

+  3.2712 

.4641 

.64 

—   .00 

I.OC 

+  .15 

•79 

.4096 

—  .6400 

•f  .6400 

+  .0960 

+  .5056 

.0000 

5.  "43 

—  .2792 

+  3-3460 

+  •7397 

'  8.9208 

14.6142 

8.9208 

[aa] 

[ac] 

[ad] 

-[an] 

[as] 

[cc] 

cd. 

—  cn. 

cs. 

dd. 

—  dn. 

ds. 

nn. 

—  ns. 

V. 

•vv. 

+  1.  0100 

+  .0909 

+  2.9088 

I.OOOO 

+  .0000 

2.8800 

.0081 

+  .2592 

+  .09 

.0081 

+  .8856 

—  .  1728 

+  1.8792 

•6724 

—  . 

1312 

1.4268 

.0256 

-.2784 

—  .07 

49 

-  .5566 

-  -4235 

—  .9075 

.2116 

1610 

•  345° 

•  1225 

+  .2625 

+  •05 

25 

+  -9555 

+  .0735 

2.  34*5 

.8281 

.  . 

0637 

2.0293 

.0049 

+  .1561 

+  -J3 

169 

•f  .9095 

-  .1284 

+  1.9902 

.7225 

—  . 

1  020 

1.5810 

.0144 

—  .2232 

—  .04 

16 

+  .6720 

—  .2400 

•3136 

—  . 

1120 

.6272 

.0400 

—  .2240 

—  -°3 

9 

—  I.  0100 

—  .0909 

—  .6161!  i.oooo 

OQOO 

.6100 

.0081 

+  .0549 

—  .  *5 

225 

—  .8288 

—  .2128 

.3020!   .5476 

!i4o6 

—  .2590 

.0361 

-.0665 

+  .02 

4 

-  .4464 

—  -2356 

1.6864 

.1296 

.0684 

.4896 

•  0361 

—  .2584 

+  .07 

49 

—  I.OOOO 

-  -3700 

—  1.0300 

i.oooo 

.3700 

I  .0300 

.1369 

-+.38" 

-1-  .12 

144 

+  .6050 

—  .0605 

+  3.4122 

.2500 

—  . 

3250 

i  .4100 

.0025 

—  .  1410 

—  .04 

16 

—  I.OOOO 

—  .1500 

—  .7900 

I.OOOO 

+  • 

1500 

.7900 

.0225 

+  .1185 

—  .10 

IOO 

+  .1958 

—  1.9201 

12.6107 

7.6754 

+  •7635 

11.9807 

•4577 

4-  .0408 

.0887 

12.6107 

11.9807 

+  .0408 

[cd] 

-[en] 

M 

[dd} 

[ds} 

[mm] 

—  l»*] 

[w] 

[vv]  =  0887. 

The  agreement  of  the  values  of  [as],  [cs],  and  [ds]  proves  the  accuracy  of  this 
part  of  the  computation. 

The  normal  equations  are  then 

5.11433  -       .'2792^  +  3.3460*  —  —     -73971 

—     .2792^  4-  14.6142*:  -f     .1958^=        1.9201; 

3.34600  +       .1958^  +  7.6754*  =  —     -7635. 


§191.  REDUCTION  OF  TRANSIT  OBSERVATIONS.  33! 


These  equations  are  now  to  be  solved,  following  the  method  and  notation  ex- 
plained in  Art.  28.  We  shall  therefore  require  the  following  auxiliary  coeffi- 
cients, viz., 

[«•!],  \cd\\  |>«i],  \cs\\  \dd-L\t  [dni]    [dsi],  [»»  i],  \ns  ij, 
[dd2\,  [dn2\,  [tts2],  [«»2],  [ns2\: 

[dsi],  \ns  i],  etc.,  being  computed  for  checks  or  the  accuracy  of  the  work. 
The  computation  will  then  be  made  according  to  the  following  scheme: 


a. 

c. 

X. 

n 

•           J. 

Proof. 

(I) 

(2> 

«3) 
U) 

(5) 
(6J 

(7> 

*a\         S.IIAT. 
log                  .70879 

[«<:!        —  -27Q? 
log          9-44592° 

[W]          3.3460 
log             .52453 

[an]  *         -  .7397 
log               9.86go6n 

[as\               8.9208 
log                d-95040 

8.9208 

log  Wi    8<737I3D 

[cc]         14.6142 
[ac]    r     , 

M  ["]  -0152 

[«*]           -1958 

teLM-.,827 

[en]              1.9201 
[^]  ta«^      -°4°4 

[«]              12.6107 

MM  -.487= 

[<rc  i]    14.599° 
log          1.16432 

[crfi]          .3783 
log         9-57807 

[<:»  i]           1.8797 
log               0.27409 

[«  i]          13-0977 
log               1.11719 

13-0978 

loggj       ,8I574 

[Af]           7.6754 
MM2,89x 

\dn\           -  .7635 
g|M.-.484o 

[ds]             11.9807 

g/M   5.8363 

Inc-  k^          «  ,  r 

[ddi]       5.4863 

^^•^ 

[</«    i]               -  -   .2795 

£3  [-]-«•, 

[ds  i]            6.1444 

C^  l]  r-c  i       r 

6.1443 

10g    [CCIJ         <J'41375 

[CC    !][«']   -3396 

:^o?2]      5.4765 
log             .73850 

[rfw  2]      —  .3282 
log               9.5i6i4n 

[^  2]                5.8048 

log                .76379 

5.8047 

log  x          8.77764.. 
.*=   -.05993 

10g  ^           9'l6°27n 

[«»]                -4577 

ra  [aw]  -io7° 

[ns]           -  .0408 
[a»]  r 

^M-X.2902 

log[^      9.10977 

[nn  i]             .3507 

|^-[C«I].2420 

[«J  ij                  1.2494 

[c*  l]  r«  ii  i  fs&L 

1-2495 

[cc  i]  [C 

loff[^l       8?7764 

[««  2]             .1087 

IX*a]rv     i 
^[^3,0x97 

[w  2]        —.437° 

[rf«a]r       , 

—  -4369 

10g    [^2]           8"  77764n 

^2][^2]       '3479 

[nn  3]             .0890 

[«*3]          -.0891 

[vv]     .0887 

PRACTICAL   ASTRONOMY.  §  19  1. 

The  accuracy  of  the  work  at  different  stages  of  progress  is  shown  by  the 
manner  in  which  the  proof  equations  are  satisfied.  Those  referred  to  by  the 
numbers  in  the  last  column  above  are  as  follows: 

(1)  (>]     =[*»]    +[«*]    +  [ad]    -[an]; 

(2)  [«  i]  =  [«•  i]  +  [«/!]-[«.  i]; 

(3)  [*  i]  -  [^  i]  +  \fd  i]  -  [</»  i]; 

(4)  [ds2-]  =  [dd2]-[dn2]; 

(5)  [«J  i]  -  [«*  i]  +  [</»  i]  -  [»»  i]; 

(6)  [ns2]  =  jy»2]  -  [«»2]; 

(7)  [ns  3]  =  -  [»«  3]  =  [H- 

We  now  determine  c  and  a  by  the  equations 

\cc  i\c  -f  \cd  i\x  —  [en  i]; 

\ad\a    -J-  [ar]c      -f-  \ad^\x  =  [an]. 

[en  i]    =         i  .  8797  [a«]      =  -     .  7397 

—  \cdi\x  =  .0227  —  \ad\x    =  -h      .2005 

_  +     1.9024  —  [ac]    c  =   +      .0364 

I4-5990 


5-H43 
1303  a  =  —     .0983 


7"!^    Weights  and  Probable  Errors. 

The  weights  of  a,  f,  and  or  will  be  given  by  formulae  (76),  viz.  : 
px=  \dd2\{ 


In  which  \dd  i] 


Therefore         /«  =    5-47°;  log  [^  i]  =  1.16432 

log  [^2]  =    .73850 


=  14.573;  \0gpo  =  I.I6354 


§  1 9 1..         REDUCTION  OF   TRANSIT  OBSERVATIONS.  333 

log  O/]2  =  8.58362 
log  p^j  =  8.83522 

Nat.  No.    .0026  7.41884 

\dd}  =  7.6754 

\dd  i]a  =  7-6728         log       *       =  9.11504 
\aa  ija 

log  p-  =  8.83522 

log  \aa\  —     .70879 

log  \cc  i]  =  1.16432 

log  [dd  2}  —     .73850 

/a  =     3-646.  log /a  =      -56187 

The  mean  error  of  a  single  observation  of  weight  unity  is — see  equation  (88)— 


In  this  case  m  —  12  ;         >u  =  3  ;         [w]  =  .0887.          Therefore  e  =  .loa 


=  %EC  =  .017  ;  Ec  =  — — r  =  .026  ; 


=  -4=  =  -052. 


We  now  have       AT  =z  3  +  x.     Therefore 

AT  —  —  6". 060  ±  .029 
c  =  -f  .130  ±  .017 
a  =  —  .098  ±  .035 

Formation  of  the  Normal  Equations  by  a   Table  of  Squares. 

We  have  seen   in  Art.  26   that  all  of  the  multiplications  necessary  for  deriv- 
ing the  normal  equations  from  the  equations  of  condition  can  be  performed  by 

*  See  equations  (27).  t  See  equations  (89). 


334 


PRACTICAL  ASTRONOMY. 


means  of  a  table  of  squares,  with  little,  if  any,  more  labor  than  by  the  use  ol 
Crelle's  table.  For  the  purpose  of  illustrating  the  method  it  will  be  applied  to 
the  present  example. 

By  referring  to  the  formulae  and  explanations  of  Art.  26  the  details   of  the 
computation  which  follow  will  be  sufficiently  clear. 


(a  +  c). 

a  +  d. 

a  —  n. 

c  +  d. 

c  —  ». 

d—n. 

aa. 

cc. 

dd. 

1.79 

1.78 

•  87 

2.01 

1.  1C 

1.09 

.6084 

.0201 

I.  OOOO 

1.08 

.82 

—  .16 

1.90 

.92 

.66 

.1664 

.6724 

—  .06 

'1.61 

1.50 

—  -75 

—  .&(. 

.81 

i.3225 

.4641 

.2116 

1.25 

i.  ii 

•  27 

1.96 

I.  12 

•98 

.0400 

.1025 

.8281 

.91 

—  .06 

1.92 

•95 

•73 

.0036 

•  I449 

•  7225 

•;$ 

.12 

—  .64 

1.76 

I.OC 

•36 

.1936 

.4400 

.3136 

-  .48 

i  53 

.62 

—   .01 

—  .92 

1.09 

.2809 

.0201 

I.  OOOO 

—  1.28 

•58 

03 

-  .38 

—  -93 

•93 

.02156 

•2544 

•  5476 

—  1.91 

—  -31 

-  :4s 

-  .88 

—  1.05 

•55 

.4489 

.5376 

.1296 

—  -34 

1.66 

1.03 

o 

-  -63 

1-37 

•4356 

.OOOO 

I.  OOOO 

+  2.37 

1.66 

i.  ii 

1.71 

i.xi 

•45 

I-3456 

.4641 

.2500 

-  .36 

1.64 

•79 

o 

-  .85 

.4096 

.0000 

I.  OOOO 

5-  "43 

14.6142 

7-6754 

[aa] 

[cc] 

[dd} 

nn. 

ss. 

(a  +  c)*. 

(a  +  d}n-. 

(a  —  n)i. 

(c  +  d}*. 

(C  -  »)2. 

(d—n)*. 

.0081 

8.2944 

3.2041 

3-1684 

•  7569 

4.0401 

I.  2100 

1.1881 

.0256 

3.0276 

1.1664 

-6724 

.0256 

3.6100 

.8464 

•4356 

.1225 

•  5625 

.0036 

2.5921 

2.2500 

•  5625 

.7396 

•  6561 

.0049 

4.9729 

1.5625 

1.2321 

.0729 

3.8416 

1-2544 

.9604 

.0144 
.0400 

3.4596 
1-2544 

1.2769 
.5776 

.8281 
.0144 

.0036 
.4096 

3-6864 
3.0976 

.9025 
I.  OOOO 

•5329 
.1296 

.0081 
.0361 

•3721 
.1225 

.2304 
1.6384 

2.3409 
•3364 

•  3844 
.0009 

.0001 
.1444 

.8464 
.8649 

1.1881 
.8649 

.0361 

1.8496 

3-6481 

.0961 

.2304 

•7744 

1.1025 

•3025 

.1369 

1.0609 

.  1156 

2.7556 

1.0609 

.3969 

1.8769 

.0025 

7-9524 

5.6169 

2.7556 

1.2321 

2.9241 

1.3456 

.2025 

.0225 

.6241 

.1296 

2.6896 

.6241 

.7225 

1.3225 

•4577 

33-5530 

19.1701 

19.4817 

7-0514 

22.6812 

11.2317 

9.6601 

19-7285 
-  .5584 

12.7897 
6.6920 

5-5720 
J-4794 

22  2896 
.3916 

15.0719 
-  3.8402 

8.I331 
1.5270 

[nn] 

H 

-  .2792 

[ac] 

£T 

•7397 
-  [an] 

•1958 

Mr 

—  1.9201 

-[en] 

-i£? 

The  proof-formula  becomes  in  this  case 
\ss\  +  2\[aa]    +  [cc]  +  {_dd}  +  [nn]\    =  [(a  +  ef\  +  [(a  +  d?}  +  [(a  -  «)a] 


which  is  completely  verified,  as  may  be  seen  by  substituting  the  above  values. 
Of  course  the  resulting  normal  equations  are  the  same  as  those  obtained  before. 


§  192.  CORRECTION  FOR  FLEXURE.  335 


Correction  for  Flexure. 

192.  The  second  form  of  transit  instrument,  that  in  which 
the  eye-piece  is  at  one  end  of  the  axis  (see  Fig.  28),  requires 
a  special  correction  for  flexure  of  the  horizontal  axis.  The 
amount  of  this  flexure  or  bending  is  assumed  to  be  the  same 
in  all  positions  of  the  telescope,  as  it  will  be  if  the  material 
of  which  the  axis  is  composed  is  homogeneous.  The  effect 
will  be  to  bring  the  reflecting  prism  lower  down  than  it 
would  be  otherwise  without  changing  the  direction  of  the 
reflecting  surface.  When  the  eye-piece  is  east  this  will 
cause  the  star  to  reach  the  collimation  axis  too  late  by  a 
small  quantity,  which  is  a  maximum  in  the  zenith  and  nothing 
in  the  horizon.  Suppose  WE  to 
represent  the  rotation  axis  bent  as 
shown  in  the  figure,  CO  being  the 
collimation  axis  of  the  telescope,  w- 
Let  E  be  the  eye  end  of  the  axis. 
The  effect  on  the  observed  time  of  vFi<T  36. 

a  star's  transit  will  evidently  be  the  same  as  that  produced 
by  elevating  the  end  marked  E,  and  when  the  proper  co- 
efficient is  found  it  may  be  combined  with  the  level  correc- 
tion. 

Lety  =  the  coefficient  of  flexure. 

/  will  be  the  maximum  displacement  of  the  transit  thread, 
and  will  be  the  value  of  this  displacement  when  the  tele- 
scope is  directed  to  the  zenith. 

The  clamp  being  on  the  end  of  the  axis  opposite  the  eye- 
piece, we  must  add  to  Mayer's  formula  the  term 


o 


cos  (<p  -  3)  (  clamp  west ) 
/'~55iT~~l  clamp  east   i" 


336  PRACTICAL  ASTRONOMY.  §  192. 

If  we  write  (g>  —  ,tf)  =  z,  the  terms  of  Mayer's  formula, 
which  give  the  correction  of  the  observed  time  of  a  star's 
transit  for  collimation,  flexure,  and  inequality  of  pivots,  may 
be  written  as  follows : 

(p  cos  2  —  /cos  z  +  c)  sec  tf;     .     .     .     (319) 

in  which/  is  determined  by  (297)  or  (297),,  and  which  we  see 
is  involved  in  the  same  manner  as  /. 

These  instruments  are  generally  provided  with  micro- 
meters, which  may  be  used  for  determining  /  and  c  at  the 
same  time,  as  follows  : 

In  order  to  make  a  satisfactory  determination,  and  at  the 
same  time  to  test  the  accuracy  of  the  assumed  law  of  change 
expressed  by  the  formula  /cos  z,  a  collimating  telescope  is 
necessary,  mounted  in  a  frame  in  such  a  manner  that  it  may 
be  placed  vertically  over  the  transit  telescope  and  at  dif- 
ferent zenith  distances  from  zero  to  90°.  The  collimation 
error  is  then  measured,  as  explained  in  Articles  182-184,  with 
the  telescope  pointed  at  various  zenith  distances.  This 
measured  value  will  include  the  term /cos  z,  which  will  be 
zero  when  z  —  90°,  and  a  maximum  when  z  —  o.  It  will 
therefore  be  possible  to  separate  c  from/. 

It  will  be  advisable  to  make  a  considerable  number  of 
measurements,  from  which  c  and /can  then  be  derived  by 
the  method  of  least  squares.  If  the  resulting  values  satisfy 
the  equations  within  the  limit  of  the  probable  error  of  meas- 
urement, the  assumed  law  of  change  expressed  by  the  for- 
mula/cos z  will  be  verified. 

In  some  cases  there  is  found  to  be  a  correction  required 
depending  on  the  temperature.  This  may  be  detected  by 
making  the  measurements  for  collimation  and  flexure  at 
different  temperatures.  If  then  different  values  are  found 
varying  with  the  temperature  according  to  any  law,  the 
necessary  correction  may  be  determined. 


CORRECTION  FOR  FLEXURE. 


337 


In  Vol.  XXXVII,  Memoirs  Royal  Astronomical  Society, 
Captain  Clarke,  R.E.,  gives  an  example  of  the  investigation 
of  the  flexure  coefficient  with  an  apparatus  of  the  kind  just 
described.  In  addition  to  the  movable  collimator,  another 
was  used  which  was  fixed  in  the  horizon.  The  collimation 
measured  on  this  was  free  from  the  effect  of  flexure,  so  that 
I y  taking  the  difference  between  the  quantity  (/  cos  z  -f  c\ 
measured  at  a  zenith  distance  z  by  means  of  the  movable 
collimator,  and  the  quantity  c,  measured  at  the  same  time  with 
the  fixed  collimator,  a  direct  measurement  of  the  quantity 
/cos  z  was  obtained.  Twelve  measurements  made  at  zenith 
distances  from  o°  to  55°  gave  the  following  results: 


z 

Difference. 

V 

z 

Difference. 

V 

z 

Difference. 

V 

O° 

2.8o 

+  22 

20° 

2.72 

+  09 

40° 

2.46 

-   15 

5 

2.68 

+  33 

25 

2.98 

-  24 

45 

I.98 

+  15 

10 

3-II 

-  13 

30 

2.40 

+  22 

50 

2.02 

—  08 

15 

3-04 

—    12 

35 

2.QO 

-  43 

55 

1.69 

+  04 

The  column  headed  z  gives  the  zenith  distance  of  the  upper 
collimator ;  the  next  column  gives  the  difference  between  the 
collimation  determined  on  the  upper  and  Iqwer  collimators; 
and  the  column  headed  v  gives  the  residuals. 

Referring  to  equation  (319),  we  see  that  the  quantity  called 
" difference"  is  equal  to  (f  —  p)  cos  z.  From  the  twelve 
measured  values  of  this  quantity  it  was  found  that 

(/—/)  =  3-°2  J  ±  -05°  expressed  in  divisions  of  the  micrometer. 
From  level-readings, 

p  —  .779  ±  .026  expressed  in  divisions  of  the  micrometer; 
therefore  ,   /=  3.800. 


PRACTICAL  ASTRONOMY.  §  194. 

One  division  of  the  micrometer  =  o".8345  ; 
therefore  /=  3".  171  —  os.2ii. 

193.  The  use  of  such  an  apparatus  as  we  have  described 
will  not  generally  be  practicable  in  the  field.  The  coefficient 
/  may  then  be  determined  from  the  observed  transits  by 
adding  to  the  equations  of  condition  (317)  the  term 

cos(?>  -  <?) 

*"  7 


cos  d 
The  complete  equation  will  then  be 

Aa  +  Bf+Cc+*T(T-  T0)  +  x  +  I  =  o.    .    (320) 


,  and  x  being  unknown  quantities.  t 

If  tiTis  known,  as  it  ordinarily  will  be,  the  number  of  un- 
known quantities  will  be  four. 

The  Transit  Instrument  out  of  the  Meridian. 

194.  Equations  (275)  and  (281)  are  strictly  general,  and  are 
applicable  to  the  reduction  of  transits  with  the  instrument 
in  any  position  whatever.  We  have  seen  that  when  the  in- 
strument is  so  near  the  meridian  that  the  squares  and  higher 
powers  of  a,  b,  m,  and  n  may  be  neglected*  these  formulae 
become  very  simple.  Bessel,  Hansen,  and  others  have  given 
more  general  methods  of  solving  the  equations  intended  for 
use  in  those  cases  where  the  observer  in  the  field  cannot  af- 
ford the  time  for  adjusting  his  instrument  accurately  in  the 
meridian.  When,  however,  the  observer  is  provided  with  a 
good  list  of  stars  reduced  to  apparent  place,  like  that  given 

*  That  is,  we  may  write  a;  bt  m,  and  n  for  sin  a,  sin  b,  etc.,  and  unity  for 
cos  a,  cos  b,  etc. 


§  IQ5-       TRANSITS  OF  THE  SUN,  MOON,  AND  PLANETS.         339 

in  the  American  Ephemeris,  this  adjustment  is  made  so 
readily,  and  the  labor  of  reduction  is  so  much  less  than  with 
the  more  general  methods,  that  the  latter  have  not  found 
much  favor,  especially  in  this  country.  Therefore,  however 
interesting  some  of  these  may  be  from  a  mathematical  point 
of  view,  we  shall  not  give  their  development  here. 


Transits  of  the  Sun,  Moon,  and  Planets. 

195.  In  the  field,  transits  of  the  moon  will  be  observed  for 
the  determination  of  longitude  when  no  better  method  is 
available.  The  sun  and  occasionally  a  planet  will  be  observed 
for  time. 

In  case  of  the  sun  and  moon  the  method  of  observing 
is  to  note  the  instant  when  the  limb  is  tangent  to  the  thread. 
With  the  sun  the  transit  of  both  limbs  may  be  observed ; 
with  the  moon  this  will  not  be  practicable  except  when  the' 
transit  is  observed  very  near  the  instant  of  full  moon.  In 
observing  a  planet,  the  transits  of  each  limb  may  be  ob- 
served alternately,  or  when  a  chronograph  is  used  both 
limbs  may  be  observed,  as  in  case  of  the  sun.  With  any  of 
these  bodies,  when  both  limbs  are  observed,  the  time  of  tran- 
sit of  the  centre  will  be  the  mean  of  that  of  the  two  limbs. 
It  may,  however,  be  desirable  to  reduce  the  limbs  sepa- 
rately for  the  purpose  of  comparison. 

When  the  moon's  limb  is  observed  on  a  side  thread,  the 
hour-angle  is  affected  by  parallax :  the  time  required  to  pass 
from  the  thread  to  the  meridian  is  affected  by  the  moon's 
motion  in  right  ascension.  The  reduction  is  as  follows: 

Let  d'  and  £  be  the  apparent  declination  and  east  hour-angle 

of  the  moon's  limb  when  observed  on  a  side  thread ; 
#  and  t,  the  geocentric  declination  and  hour-angle ; 
z  and  z' ,  the  geocentric  and  apparent  zenith  distance. 


340  PRACTICAL   ASTRONOMY.  §  19$. 

We  can  reduce  the  observation  by  either  of  the  equations 
(282),  (283),  or  (284).  Taking  the  latter,  viz.,  Mayer's  for- 
mula, we  have  ^^  J 

sin  (9  -  *>)        cos  (cp  -  (T)      cf  +  i 
-~~-~—  "      (32I) 


i  being  the  equatorial  interval  of  the  thread. 

Having  /',  /  may  be  determined  as  follows: 

In  Fig.  37,  let  P  be  the  pole,  Z  the  zenith,  O  the  geocentric 
place  of  the  moon  at  the  instant  of  observation,  O-  the  ap- 
parent place. 

Angle  MPO  =  t ;        ZO  =  z ; 
MPO'  =  /';        ZO  =  z'. 

From  the  triangles  MZO  and  M'ZO', 

sin  MO       sin  M'O' 

sin  MZO  =  — = =  — = / — .     .     (322) 

sin  z  sm*' 

J — -VFrom  triangle  MPO,       sin  J/0  =  sin  t  cos  £ ;   )  /      x 
FIG.  37.    From  triangle  M'PO',  sin  J/'^7  =  sin  t'  cos  tfr.  I 


Substituting  these  values  in  (322),  we  have 

sin  /  cos  3       sin  t'  cos  8' 
~~sin  -sr  sin  ^' 


.  cos  <^r     sin  z  .      x 

As /is  small,  /  =/          - .          r,    .    .    .    (324) 


the  required  value  of  t  in  terms  of  t'. 
•  Let  A  =  the  increase  of  the  moon's  right  ascension  in  one 


TRANSITS  OF   THE  MOON. 


341 


sidereal  second ;  then  /  being  expressed  in  seconds,  the  time 
required  for  the  moon  to  pass  over  this  interval  will  be 


i- A' 


(325) 


i  —  A  representing  the  velocity  with  which  the  moon  ap- 
proaches the  meridian.  * 

There  remains  the  correction  for  the  moon's  semidiameter. 

Let  5  =  the  geocentric  semidiameter  of  the  moon  at  the 

time  of  transit,  taken  from  the  ephemeris; 
S'  =  the  hour-angle  of  the  centre  when  the  limb  is  on 
the  meridian.  p 


Then,  from  Fig.  38, 


sin  S'  = 


sin_5 
cos~<T 


Writing  5  and  Sf  for  their  sines  and  dividing  by  15 
to  reduce  to  time, 


15  cos  <r 


FIG.  38. 


The  time  required   for  the  moon    to  pass  over  this  space 
will  be 


S' 


l  -  A 


15(1  —  A)  cos 


(326) 


From  (321),  (324),  (325),  and  (326),  we  have  for  the  right 
ascension  of  the  moon's  centre  when  the  limb  is  observed  on 
any  thread  of  the  transit  instrument, 


I5(i-A)cos5 


,       . 
~-       (327) 


342  PRACTICAL  ASTRONOMY.  §  195. 

The  geocentric  declination,  #,  and  the  equatorial  horizontal 
parallax,  n,  are  taken  from  the  ephemeris.  Then  from  (XI)X, 
Art.  85,  we  have  with  sufficient  accuracy  for  this  purpose 

df  =  d  —  np  sin  (<pf  —  <T);    ....     (328) 

where  generally  d  may  be  substituted  for  df,  and  cp  for  <p',  in 
the  second  member.  % 

Then/  being  the  parallax  in  zenith  distance,  we  have 


and  the  factor  —. -,  in  equation  (327)  becomes 

sin  z 

sin  z  sin  z 

— 7  =  — —  — ; : — -  =  cos  /  —  cot  z  sin  / 

sin  z        sin  z  cos/  -)-  cos  z  sin/ 

approximately.    And  from  (VII),,  Art.  82   with  sufficient  ac- 
curacy for  this  purpose, 

sin  z  . 

— — -  =  i  —  p  sin  n  cos  (or  —  tf). 

sins' 

If  then  we  write    Al  =  i  —  p  sin  n  cos  (cpr  —  (5), 

i 

F  =  ^4^  sec  tf, 


(329) 


^j  may  be  tabulated  with  the  argument  log  p  sin  TT  cos(^>/— tf) 
as  in  table  XIII  of  Bessel's  Tabttlce  Regiomontance ;  B,  may 
be  tabulated  with  the  argument  Act  —  moon's  change  in  right 

ascension  in  one  minute,  Aa  being  given  in  the  ephemeris. 
£ 

The  term  -  — ^  may  be  taken  from  the  table   of 

15  (i  —  A)  cos  d 

"  Moon  Culminations"  of  the  ephemeris  where  it  is  given 
under  the  heading  "  Sidereal  time  of  semidiameter  passing 


§  196.  MOON'S  DEFECTIVE  LIMB.  343 

the  meridian."     The  complete  formulae  for  the  moon's  right 
ascension  are  then  as  follows  : 

d'  =  d  —  np  sin  (<pr  —  tf); 
Al  =  I  —  p  sin  n  cos  (<p'  —  tf); 


=  AlBl  sec  tf; 

5')  .    _cos(<f>— £')          c'    \  S 

f-  <? T7 r.  \  .T  COS  O  + 

cos  6'  cosS'J  15(1  —  X)  cos  < 

The  use  which  will  be  made  of  this  value  of  a  in  the  de- 
termination of  longitude  will  be  explained  hereafter.  A 
series  of  stars  will  be  observed  in  connection  with  the  moon 
for  determining  the  clock  correction  A  T  and  the  constants 
a  and  c.  Sometimes  the  clock  correction  is  made  to  depend 
exclusively  on  about  four  stars  whose  declination  is  nearly 
the  same  as  that  of  the  moon ;  two  of  these  precede  the  moon 
and  two  follow. 

Correction  to  the  Moon's  Defective  Limb. 

196.  The  transit  of  both  limbs  of  the  moon  can  only  be 
observed  when  the  culmination  occurs  very  near  the  time  of 
full  moon.  If  one  limb  is  defective  it  may  still  be  used  if  it 
is  sharply  defined,  and  a  correction  applied  for  defective 
illumination. 

For  this  purpose  we  may  regard  the  moon  as  a  sphere,  and 
we  may  consider  the  rays  of  light  from  the  sun  to  the  moon 
as  parallel  to  those  from  the  sun  to  the  earth.  The  curve  of 
contact  of  the  surface  of  the  moon  with  the  cone  of  rays  tan- 
gent to  its  surface  will  separate  the  light  from  the  dark  part 
of  the  moon.  When  the  defective  limb  is  observed,  the 
point  whose  contact  with  the  thread  of  the  reticule  is  noted 
is  a  point  on  this  curve;  and  instead  of  the  semidiameter  5, 


344 


PRACTICAL  ASTRONOMY, 


196. 


we  shah  require  for  this  limb  the  perpendicular  from  the 
centre  of  the  disk  upon  the  hour-circle,  of 
which  the  transit  thread  may  be  regarded 
as  forming  a  small  arc.  Thus  aa'  being  the 
position  of  the  thread  at  the  instant  of  the 
observed  transit  of  the  defective  limb  Z, 
we  shall  require  the  distance  CL  =  S1  in- 
stead of  S.  Fig.  40  may  be  regarded  as 
a  section  formed  by  the  plane  passing 
FIG.  39.  through  the  rotation  and  collimation  axes 

of  the  instrument,  and  Fig.  39  a  section  formed  by  the  plane 
perpendicular  to  the  collimation  axis. 

E  is  the  point  on  the  earth's  surface  from  which  the  obser- 
vation is  made. 

E~2  and  K2  are  the  projections  on  the 
plane  of  the  instrument  of  rays  of  light 
coming  from  the  sun.  .These  lines  are 
practically  parallel. 

Let  x  =  the  angle  formed  with  the  plane 
of  the  meridian  by  the  line 
drawn  from  the  sun  to  the 
moon. 


This  will  be  practically  the  same  angle 
as  that  formed  by  lines  joining  the  sun  and 
earth. 

CK  will  be  perpendicular  to  this  line. 
Also,  KN  is  perpendicular  to  the  plane  of 
the  meridian.  Therefore 


Sl  —  S  cos  x. 


(330) 


FlG  40  x  is  now  the  angle  which  a  line  drawn 

from  the  sun  to  the  earth  forms  with  the  lower  branch  of  the 
meridian. 


§  196.  TRANSIT  OF   THE  MOON.  345 

Let  a  =  the  moon's  right  ascension  at  the  time  of 

culmination ; 

a'  =  the  sun's  right  ascension  ; 
a'  —  a  =  angle  formed  by  the  hour-circles  drawn 

through  the  moon  and  sun ; 
1 80°  —  (of  —  a)  =  angle  formed  by  sun's  hour-circle  with 

the  lower  branch  of  the  meridian. 
6f  —  sun's  declination. 

In  Fig.  41,  E  is  the  earth,  Pihe  pole  of  the  heavens,  and  5 
the  projection  of  the  sun  on  the  celestial 
sphere.     PR  is  the  lower  branch  of  the 
meridian.     SR  is  the  arc  of  a  great  circle 
perpendicular  to  the  meridian. 

Therefore  SER  =  x  =  arc  SR. 

The  right-angle  triangle  SPR  there- 
fore gives  FlG  4I* 

sin  x  =  cos  <$'  sin  (a'  —  ex) (33 1) 

(330)  and  (331)  therefore  give  the  required  value  of  S',  and 
the  correction  to  be  applied  will  be  of  the  same  form  as  in 

case  of  5,  viz.,  ±  — 7 YT •=  v      >  when  <  •  jl  limb 

"  15  (i  —  A)  cos  #  (  —  j  (  second  j 

is  defective. 

Example.     1883.  October  15,  the  moon  was  observed  with  the  portable  transit 
instrument  of  the  Sayre  observatory  as  follows: 

Cl.  east. 


First  Limb.  ' 

Second  Limb. 

Level. 

I 

2I8.2 

i 

43s-  1 

E. 

w. 

II 

38.8 

ii 

0  .1 

12.9 

12.9 

III 

55-1 

in 

16  .9 

ii.  5 

14.2 

IV 

12.5 

IV 

34 

12.9 

12.9 

V  i" 

i6m  29s 

V  ih 

i8m  so8.  8 

II-  3 

14.4 

T—\    15     55.32  i   18     16.98  12.15       13.60 


346  PRACTICAL  ASTRONOMY.  §  196. 

From  the  table  of  moon  culminations  (page  379  of  the  ephemeris)  we  find,  for 
the  time  of  the  moon's  transit  at  Bethlehem: 

Apparent  declination  =  S  =  9°  14'  18" 

Equatorial  horizontal  parallax  =  it  =  3681" 

A  .0425 

Sidereal  time  of  semidiameter  passing  the  meridian  =  7o8.76 

We  also  have  q>'  —      40°  25'    2" 

log/o=  9-99939 

Correction  for  inequality  of  pivots  =  p  —  —        .062 

The  computation  by  formulae  (XIX),  Art.  195,  is  now  as  follows: 

<p'  —  d  =  31°  10'  44"        sin  (q>  —  6)  =  9.7141        cos  (<p'  —  d)  =  9.9323 

log  7t  •=.  3.5660  sin  7t  =  8.2515 

log  p  =  9.9994  log  p  =  9.9994 


Sum  =  3.2795  Sum  =  8.1832 

Nat  No.  1903"  .01525 

S'  =    8°  42'  35"  Al         .98475 

i  -  A.  =  0.9575 
log(i  —  A)  =  9.9811 

log  B!   =      .0189 

cos  d'  =  9.9950 
log  F  —   ".0179 
log  /^cos  d'  =    .0129  F  cos  S'  •=.  1.030 

The  above  level-readings  in  connection  with/  give     b  —  -f-  ".115. 
We  have  derived  from  transits  of  stars  c'  =  -f~  •I54'» 

a  =  -   .065; 
AT-  -  5'.47. 

We  now  apply  the  last  of  formulae  (XIX): 


cos<F=    ^'56 
sum  =  -}-  .220 
(Sum)  .Fcos  d'  =  -f  '.227 


§  197.  TRANSIT  OF  THE  SUN  AND  PLANETS.  347 

First  Limb.  Second  Limb. 

T  =  ih  I5m  558.32  ih  i8m  i68.98 

AT-          -    5.47  -    5-47 

Corrections               +       -23  +      -23 

Right  ascension  of  limb        ih  I5m  5O9.o8  ih  i8m  n*.74 

The  right  ascension  of  the  centre  will  be  obtained  from  either  of  these  by 
applying  the  correction  for  semidiameter,  which  is  the  same  as  the  sidereal  time 
of  the  semidiameter passing  the  meridian.  The  illumination  of  the  second  limb, 
however,  was  defective,  and  therefore  the  correction  given  by  formulae  (330)  and 
(331)  should  be  applied. 

From  the  ephemeris  we  have* 

Sun's  right  ascension       =  a'  =      I3h  23m  io8 
Sun's  declination  =  d'  =  —  8°  45'  18" 

Moon's  right  ascension  =  a  =       ih  I7m    i8 

Applying  formula  (331),  a'  —  a  —      i2h    6m    9" 

=    181°  32'  15" 

sin  (a'  —  a)  =  8.4286 
cos  d'  =  9.9949 
sin  x  —  8.4235 
cos  x  -  9.99985 
log  70".  76  =  1.84979 
Corrected  value  =  70^.74  log  =  1.84964 

Therefore 

Right  ascension  moon's  centre  from  observation  of  first  limb  =  ih  I7mo8.84. 
Right  ascension  moon's  centre  from  observation  of  second  limb  =  I    17     1 .00. 


Transits  of  the  Sun  and  Planets. 

197.  Formulas  (XIX)  derived  for  the  moon  apply  equally 
to  the  sun  and  planets.  As,  however,  the  parallax  in  these 
cases  will  always  be  small,  we  can  write  without  appreciable 
error  z  =  z'  and  S  —  8'. 


Then        A,  =  i;        Bl  =  —  —^\        F  =  B,  sec 

I   —  A 


5(1—  X)  cos  S 


(33.) 


PRACTICAL   ASTRONOMY.  §  198. 

The  last  term  can  be  taken  directly  from  the  ephemeris, 
where  it  is  given  under  the  heading  "  Sidereal  time  of  semi- 
diameter  passing  the  meridian."  The  object  of  such  an  ob- 
servation will  be  to  determine  the  clock  correction  AT. 

If  the  sun  is  observed  with  a  mean  time  chronometer,  the 
rate  of  which  is  small,  A  may  be  neglected,  as  then  the  motion 
of  the  sun  will  practically  correspond  with  that  of  the  chro- 
nometer. If  the  chronometer  has  a  large  rate  on  apparent 
time,  this  rate  may  be  placed  equal  to  A,  -f-  when  the  chro- 
nometer is  gaining,  —  when  losing. 

Let  E  =  the   equation   of  time   for   the   instant  of 

transit  ; 
S"  =  the  mean  time  of  semidiameter  passing  the 

meridian  ; 
T  =  chronometer  time  of  observation  reduced 

to  middle  (or  mean)  thread  ; 

AT  =  the  chronometer  correction  on  mean  time. 
Then  I2h  +  E  =.  mean  time  of  sun's  transit. 

Therefore 


S"  is  -f-  for  preceding  limb,  and  —  for  following  limb  ; 
when  both  are  observed  it  vanishes  from  the  mean.  AT 
will  then  be  given  by  (333). 

The  'Transit  Instrument  in  the  Prime  Vertical. 

198.  The  transit  may  be  employed  for  determining  the  in- 
stant of  a  star's  passing  the  prime  vertical,  in  a  manner  simi- 
lar to  that  already  explained  for  determining  its  passage  over 
the  meridian.  Such  observations  furnish  a  very  accurate 
method  of  determining  the  latitude  of  the  place  of  observa- 


198. 


PRIME    VERTICAL    TRANSITS. 


349 


tion,  or,  in  a  fixed  observatory  where  the  latitude  is  known, 
for  determining  the  declinations  of  the  stars  observed.  The 
practical  application  of  the  transit  to  these  purposes  is  due 
to  Bessel,  although  a  prime  vertical  transit  was  used  by 
Roemer  more  than  a  hundred  years  earlier. 

This  method  of  determining  latitude  has  been  considerably 
used  by  the  astronomers  of  Europe,  arid  to  a  less  extent  in 
America.  It  is  now  almost  en-  P 

tirely  superseded  by  the  use  of 
the  zenith  telescope,  so  that  a 
complete  presentation  of  the 
theory  is  relatively  much  less  im- 
portant now  than  it  was  thirty  or 
forty  years  ago. 

The  principle  is  as  follows :  Let  ~ 
P  be  the  pole,  Z  the  zenith,  and  .S 
a  star  which  crosses    the  prime  FIG.  42. 

vertical  at  5  and  S' .  Suppose  the  instant  of  the  star's  passing 
the  prime  vertical  to  be  observed  with  a  transit  instrument 
perfectly  adjusted  in  this  plane ;  then  if  the  rate  of  the  clock 
is  known,  the  difference  between  the  two  times  of  transit 
will  be  the  angle  SPS',  one  half  of  which  is  equal  to  SPZ  =  t. 
Then  from  the  right-angle  triangle  SPZ  or  SfPZ  we  have 


tan  cp  =  tan  #  sec 


(334) 


from  which  either  cp  or  d  may  be  determined  when  the  other 
is  known.  In  the  field  it  will  of  course  be  cp  which  is  to  be 
determined. 

The  process  is  then  analogous  to  that  employed  with  the 
instrument  mounted  in  the  meridian ;  viz.,  the  adjustments 
are  made  as  accurately  as  may  be,  and  the  corrections  to  the 
final  result  determined  for  outstanding  deviations.  As  we 
shall  see,  the  value  of  the  method  consists  largely  in  the 


350  PRACTICAL   ASTRONOMY.  §  199. 

facility  with  which  the  effect  of  instrumental  errors  may  be 
eliminated.  It  is  evident  that  only  those  stars  can  be  ob- 
served on  the  prime  vertical  which  culminate  between  the 
equator  and  the  zenith,  that  is,  whose  declinations  are  be- 
tween o  and  <p. 

Adjustments. 

199.  It  is  only  necessary  to  explain  the  method  of  placing 
the  instrument  in  the  prime  vertical,  all  the  remaining  ad- 
justments being  the  same  as  when  the  instrument  is  in  the 
meridian.  For  this  purpose  a  star  is  selected  whose  declina- 
tion is  small,  and  the  clock  time  computed  when  the  star  will 
be  on  the  prime  vertical.  Triangle  PSZ  of  Fig.  42  gives 

tan  6 


The  clock  time  of  the  star's  passing  the  prime  vertical  will 
then  be 


(336) 


When  the  clock  time  is  that  given  by  this  formula,  the 
middle  thread  of  the  reticule  must  be  brought  on  the  star  by 
the  fine-motion  azimuth  screw. 

It  will  be  observed  that  a  knowledge  of  the  latitude  is 
necessary  for  computing  /,  but  from  (335)  it  appears  that 
when  a  star  is  chosen  whose  declination  is  nearly  o,  a  small 
error  in  the  assumed  value  of  <p  may  exist  without  mate- 
rially affecting  the  value  of  /.  The  adjustment  should  be 
tested  by  stars  both  east  and  west  of  the  meridian,  as  an 
error  in  the  assumed  value  of  <p  will  affect  the  computed 
times  for  east  and  west  stars  with  opposite  signs. 


§  2OO.  PRIME    VERTICAL    TRANSITS.  351 

Some  instruments  are  provided  with  azimuth  circles  like 
that  shown  in  Fig.  28,  in  which  case  the  simplest  method  of 
proceeding  will  be  to  first  adjust  the  instrument  in  the  plane 
of  the  meridian  and  then  turn  it  in  azimuth  90°  by  the  circle. 


Method  of  Observing. 

200.  A  list  of  stars  to  be  observed  should  first  be  pre- 
pared, for  which  the  time  of  passing  the  prime  vertical,  both 
east  and  west,  must  be  computed,  also  the  zenith  distance  or 
setting  of  the  finding  circle.  Formulas  (335)  and  (336)  give 
the  required  time.  The  zenith  distance  is  given  by 

sin  8  t      \ 

cos  2  =  -. .     (337) 

sin  cp 

If  the  star  is  near  the  zenith,  the  time  required  to  pass  the 
thread  intervals  will  be  comparatively  large,  so  that  it  will 
be  convenient  to  compute  approximately  the  time  of  passing 
the  first  thread. 

Let  i  —  the  equatorial  interval  of  the  first  thread ; 
/  —  the  corresponding  star  interval. 

Then  /  =  — ^^ —  =  - : approximately.  (338) 

sin  cp  cos  d  sin  t       sin  (p  sin  z   J 

The  proof  of  this  formula  will  be  given  hereafter.  /  will 
be  subtracted  from  the  time  given  by  (336)  for  a  star  either 
east  or  west. 

As  the  star  moves  obliquely  across  the  field,  it  will  be 
necessary  to  change  the  zenith  distance  of  the  telescope  for 
every  thread  in  order  to  have  the  transits  take  place  between 
the  two  horizontal  threads. 


352  PRACTICAL   ASTRONOMY.  §  2OI. 

Mathematical  Theory. 

201.  The  equations  (275)  and  (281)  apply  to  the  transit  in- 
strument in  any  position  whatever,  and  consequently  may  be 
used  in  this  case.  It  will  perhaps  be  better  to  derive  the 
formulae  directly. 

Let  us  consider  the  point  where  the  north  end  of  the  axis 
produced  pierces  the  celestial  sphere.  This  we  shall  call  the 
north  end  of  axis. 

Let  this  point  be  referred  to  a  system  of  rectangular  axes, 
the  horizon  being  the  plane  of  xy,  the  positive  axis  of  x  being 
directed  north,  the  positive  axis  of  y  east,  and  the  positive 
axis  of  z  to  the  zenith. 

Let  a  =  the  azimuth  of  the  north  end  of  axis,  reckoned 

from  the  north  point  towards  the  east ; 
b  =  the  altitude. 

Then  x  =  cos  b  cos  a\        y  =  cos  b  sin  a ;        z  —  sin  b.   (339) 

In  the  second  system  let  the  equator  be  the  plane  of  xy, 
the  positive  axis  of  z  being  parallel  to  the  earth's  axis,  the 
positive  axis  of  x  being  directed  to  the  point  where  the  lower 
branch  of  the  meridian  intersects  the  equator,  and  the  axis 
of  y  coinciding  with  that  in  the  first  system. 

Let  n  and  180°  +  ^  =  tne  declination  and  hour-angle  of 

the  north  end  of  axis. 

Then   x'  =  cos  n  cos  m\    y'  =  cos  n  sin  m]     z'  —  sin  n.  (340) 
The  formulae  for  transformation  of  co-ordinates  give 

cos  n  cos  m  —  cos  b  cos  a  sin  cp  —  sin  b  cos  y,  ) 
cos  n  sin  m  =  cos  b  sin  a  ;  >•     (341) 

sin  n  =  cos  b  cos  a  cos  q>  +  sin  b  sin  cp.  ) 


§  202.  PRIME    VERTICAL    TRANSITS.  353 

If  the  instrument  is  carefully  levelled  and  adjusted  in  the 
prime  vertical,  we  way  write 

cos  b  =  i  ;        cos  a  —  i  ;        sin  b  =  b  ;        sin  a  =  a  ; 
when  the  above  equations  may  be  written 

cos  n  cos  m  =  sin  (q>  —  b)  ;  j 
cos  w  sin  ;/z  =  #  ;  V  ....     (342) 

sin  ^  =  cos  (<p  —  b).  } 

We  shall  find  these  formulae  useful  in  subsequent  transfor- 
mations. , 

202.  Let  90°  -f-  c  =  the  angle  between  the  clamp  end  of 

the  rotation  axis  and  the  object  end 
of  the  collimation  axis  ; 

t  and  d  =  the   hour-angle  and  declination   of  .a 
star  observed  on  the  middle  thread. 

Let  the  star  be  referred  to  a  system  of  rectangular  axes, 
the  equator  being  the  plane  of  xy,  the'axis  of  x  being  directed 
to  the  point  where  the  hour  circle  through  the  north  end  of 
the  rotation  axis  intersects  the  equator. 

Then  the  angle  formed  by  the  radius  vector  with  the  plane 
of  xy  will  be  d,  and  the  angle  between  the  projection  of  the 
radius  on  the  plane  of  xy  and  the  axis  of  x  will  be 

1  80°  +  (/.-  m). 


x  =  —  cos  tfcos(/—  m)\  y  —  —cos  tf  sin  (t—m)\  z  =  sin  rf.  (343) 

In  the  second  system,  let  the  axis  of  x  coincide  with  the 
rotation  axis,  the  axis  of  y  coinciding  with  that  of  the  former 
system.  Then  the  position  of  the  instrument  being  clamp 
north,  —  c  will  be  the  angle  formed  by  the  radius  vector  and 


354  PRACTICAL   ASTRONOMY.  §  203. 

the  plane  of  yz.  Let  ^  be  the  angle  formed  with  the  axis  of 
y  by  the  projection  of  the  radius  vector  on  the  plane  of  yz. 
Then 

x'  =  —  sin  c  ;      y'  —  cos  c  cos  #,  ;      <sr'  =  cos  £  sin  £,.  (344) 
The  angles  between  the  axis  of  x  and  ^  being  n,  we  have 
.£•'=:  xcosn-{-  ^sin«;    y=^;     #'1=  —  .rsin  n  -f-  #costf.  (345) 
We  therefore  have 

sin  £  =       cos  $  cos  (V  —  m)  cos  #  —  sin  d  sin  «;  j 
cos  c  cos  tf,  —  —  cos  #  sin  (/  —  m)\  V  (346) 

cos  £  sin  #,  =       cos  d  cos  (t  —  m)  sin  n  -f-  sin  (J  cos  ».  ) 

.  Equations  (341)  and  (346)   express   in  the  most  general 

form  the  relations  between  the  quantities  which  determine 

the  position  of  the  instrument  and  the  quantities  <p,  6",  and  /. 

203.  The  adjustments   may  always   be  made   accurately 

enough  so  that  the  first"  of  (346)  may  be  written 


c  =  cos  d  sin  (9-b)-  sin  d  cos  (<p  -  t);    (347) 

COS  tri 


where  the  values  of  sin  n  and  cos  n  given  by  (342)  have  been 
substituted. 

Let  h  sin  cpf  =  sin  d  ; 

cos  tf  cos  (t  —  m) 
' 


.     0, 
.....    (343) 

Then  (347)  becomes      c  —  h  sin  (<p  —  cpf  —  b). 


§  2O4-  PRIME    VERTICAL    TRANSITS.  355 

From  the  first  of  (348),  h  =  -^  —  7,  and  therefore  when  tf  is 
not  too  small  we  may  write 

sin  O  -  <p'  -  b)  =  <p  -  <p'  -  b  =  ~- 


.    c  sn  #/ 
or  <p  =  9  +  £  +  -g^--       ....     (349) 

Dividing  the  first  of  (348)  by  the  second,  we  obtain 

tan  cp'  —  tan  d  sec  (t  —  m)  cos  m.     .     .     .     (350) 

When  c,  m,  and  £  are  known  quantities,  (349)  and  (350) 
will  give  the  latitude,  as  tf  is  the  known  declination  of  the 
star,  and  /  is  obtained  by  observation. 

204.  b  is  determined  as  in  previous  discussions  by  the 
striding-level.  This  should  be  done  with  care,  as  we  see  from 
(349)  that  an  error  in  b  will  affect  the  latitude  by  its  full 
amount,  t  and  m  are  determined  as  follows: 

Let  /'  and  t  =  hour-angles  of  the  star  at  east  and  west  tran- 

sit respectively  ; 
T  and  T  =  observed  clock  times  at  east  and  west  tran- 

sit respectively  ; 
A  T  and  A  T  =  corresponding  clock  corrections  ; 

23-  =  elapsed  time  between  east  and  west   ob- 

servation ; 

a  =  star's  right  ascension  =  sidereal  time  of  cul- 
mination. 

Then  t'  =  T  +  AT  -  a-, 

t  =  T  +  A  T  -  a; 


a.      .      (351) 

Therefore     3  =  /  —  m  =  —  (f  —  m}. 


PRACTICAL   ASTRONOMY.  §  205. 

For  determining  3  we  see  that  the  clock  rate  must  be 
known,  but  neither  the  clock  correction  nor  the  star's  right 
ascension  is  required.  For  determining  m  a  knowledge  of 
both  these  quantities  will  be  essential. 

With  the  portable  instrument  c  may  most  readily  be 
determined  by  observation  in  the  meridian,  as  already  ex- 
plained,* but  on  account  of  the  facility  with  which  an  error 
in  this  quantity  may  be  eliminated  its  exact  determination  is 
not  very  important. 


Effects  of  Errors  in  the  Data. 

205.  Let  us  now  investigate  the  effect  upon  the  latitude  of 
uncorrected  errors  in  the  quantities  b,  c,  tf,  and  $  —  t  —  m. 

Suppose  the  same  star  observed  both  east  and  west  on  two 
different  nights,  first  with  the  instrument  in  the  position 
clamp  north;  second,  clamp  south. 

Let  b  and  b'  —  the  inclination  given  by  the  level  for  clamp 

north  and  south  ; 
p  =  the  (unknown)  correction  for  inequality  of 

pivots ; 

c  •=.  collimation  constant,  +  f°r  damp  north  ; 
q  =  the  unknown  error  in  determining  c. 

Then  (b  +/)  and  (b'  —  p)  —  the  true  inclination  of  axis  for 

clamp  north  and  south  respec- 
tively ; 

c  +  q  =  true  value  of  collimation  con- 
stant. 


See  equation  (305). 


§205-  PRIME    VERTICAL    TRANSITS.  357 

Let  (pf  and  q>"  =  the  latitude  given  by  (350)  from  transits 
of  the  same  star  clamp  north  and  south 
respectively. 
Then  (349)  gives 

9  =  <P'  +  b  +/  +  (c  +  4)  -gjfg-  clamp  north; 

cp  =  cp"  -f  b'  —  p  —  (c  -f-  q)   s     Q  clamp  south. 

The  mean  is 

n  ' *".  (352) 


Unless  the  errors  of  adjustment  are  very  large  the  last 
term  of  this  equation  will  be  inappreciable,  so  that  practi- 
cally constant  errors  of  collimation  and  level  are  eliminated 
by  combining  observations  on  the  same  star  in  different 
positions  of  the  axis. 

Errors  in  5  may  result  either  from  errors  in  the  clock  rate 
or  they  may  be  simply  the  unavoidable  errors  of  observation. 
To  ascertain  their  effect  upon  cp  we  differentiate  (350)  with 
respect  to  cp  and  S*,  by  which  means  we  derive 

dtp  =  isin  2cp  tan  3  d$  (nearly).     .     .     .     (353) 

From  this  equation  it  appears  that  an  error  in  3  will  produce 
the  less  effect  upon  cp  the  smaller  3  is.  Also,  that  the  alge- 
braic  sign  when  the  star  is  east  is  the  opposite  of  that  when 
it  is  west.  Therefore 

The  effect  of  a  small  error  in  3  will  be  eliminated  by  ob- 
serving the  star  both  east  and  west  of  the  meridian. 

Differentiating  (350)  with  respect  to  cp  and  tf,  we  find 

sin  2cp 

^  =          (354) 


35$  PRACTICAL  ASTRONOMY.  §  2o6. 

As  the  declination  cannot  be  greater  than  <p,  we  see  that 
when  cp  is  less  than  45°  an  error  in  #  will  produce  a  larger 
error  in  <p.  For  g>  greater  than  45°,  dq>  <  dd  for  all  stars 
whose  d  is  between  q>  and  90°  -  -  q>.  In  any  case  the  effect 
upon  <p  will  be  less  the  nearer  the  star  is  to  the  zenith. 

The  best  result  will  therefore  be  obtained  by  observing  at 
both  the  east  and  west  transit  a  star  which  culminates  near 
the  zenith  and  in  both  positions  of  the  axis.  The  observa- 
tions may  be  made  on  the  same  star  on  two  different  nights, 
the  clamp  being  north  in  one  case  and  south  in  the  other. 
Or  they  may  all  be  made  on  the  same  night  if  the  star  passes 
quite  near  the  zenith,  as  follows :  First,  observe  the  east 
transit  over  the  first  half  of  the  threads  of  the  reticule; 
second,  reverse  the  instrument  and  observe  the  transit  over 
the  same  threads,  now  in  the  reverse  position  ;  third,  ob- 
serve the  west  transit  over  the  same  threads ;  then,  fourth, 
reverse  the  instrument  again  and  finish  the  observation  of 
the  west  transit  over  the  threads,  now  in  the  same  position 
as  at  first.  This  method  is  due  to  Struve.  It  will  not  gener- 
ally be  followed  in  the  field  owing  to  the  danger  of  disturb- 
ing the  instrument  in  reversing  so  frequently. 

Reduction  to  the  Middle  or  Mean  Thread. 

206.  In  formula  (349),  c  is  the  error  of  collimation  of  the 
middle  or  mean  thread.  In  reducing  the  observations  over 
a  side  thread  we  may  replace  c  by  c  -f  i  (i  being  the  equa- 
torial interval  of  the  thread),  and  reduce  each  thread  sepa- 
rately. It  will,  however,  be  simpler  to  first  reduce  all  obser- 
vations to  the  times  over  the  middle  or  mean  threads.  This 
process  is  less  simple  than  in  case  of  meridian  observations, 
since  the  mean  of  the  times  over  the  several  threads  will  not 
in  this  case  be  the  time  over  the  mean  thread. 

The  reduction  may  be  made  in  either  of  two  ways :  first, 


§  206.  REDUCTION   TO  MIDDLE    THREAD.  359 

by  reducing  each  thread  separately  to  the  middle  (or  mean) 
thread  ;  second,  by  applying  a  correction  to  the  mean  of  the 
times  over  the  different  threads  to  reduce  it  to  the  time  over 
the  mean  thread. 

v  First.  The  thread  intervals  should  be  determined  by  meri- 
dian transits  as  already  explained.* 

Let  i  —  the  equatorial   interval  of  any  thread  from  the 

middle  thread  ; 

/  =  the  corresponding  star  interval  ; 
t  •=.  the  hour-angle  of  the  star  when  on  the  middle  (or 

mean)  thread  ; 

t  —  I  =  the  hour-angle  when  on  the  side  thread  ; 
c  -f-  i  may  be  regarded  as  the  collimation  error  of  the  side 
thread. 

Then,  from  the  first  of  (346), 

sin  (c  -f  z  )  =:  —  sin  n  sin  $  +  cos  n  cos  8  cos  (t  —  I  —  m)  ; 
sin  c  =  —  sin  n  sin  8  -\-  cos  n  cos  8  cos  (t  —  m). 

Subtracting,  we  readily  find 
2  cos  (J-*  -j-  c)  sin  \i  =  cos  n  cos  $2  sin  \t  —  m  —  £7)  sin  f/. 

Since  c  will  be  very  small,  the  first  term  of  this  may  be  writ- 
ten sin  i  without  appreciable  error.  Then 

.    j_  _  sin  i 

~ 


cos  n  cos  fi  sin  (/  -  m  - 

From   (342)   we   may  write   cos  n  =  sin  (cp  —  b}.      Also, 
(/  —  m)  =  S.     sin  i  may  be  written  i. 


2  sin  \I  =  7(i  -  ^7°)  =  7(cos  /)*. 

*  Art.  174. 


360  PRACTICAL  ASTRONOMY.  §  207. 

Therefore  (355)  may  be  written  without  appreciable  error, 


~    sin  (cp  —  b)  cos  8  sin  (3-  —  £/)  (cos  7) 
and  with  accuracy  sufficient  for  most  cases, 

==  sin  <p  cos  tf  sin  (£ 

Log  (cos  7)A  might  be  tabulated,  but  it  will  be  required 
so  rarely  that  it  will  hardly  repay  the  labor.  The  value  of 
7  required  in  the  second  member  of  the  above  formulae 
may  be  found  directly  from  the  observations  themselves,  by 
taking  the  difference  of  the  observed  time  over  the  side 
thread  and  middle  thread. 

*  Care  must  be  taken  to  give  the  proper  algebraic  signs  to 
i,  7,  and  $,  —  i  and  7  being  plus  for  north  threads  and  minus 
for  south  ones  ;  5,  plus  for  west,  minus  for  east  transits. 

207.  Second.  This  method  of  reduction  is  due  to  Bessel, 
and  is  more  convenient  when  many  stars  are  to  be  reduced. 
Resuming  the  first  of  (346),  and  writing  c  +  i  instead  of  sin  c 
and  t  —  I  for  /, 

c  +  i  —  —  sin  n  sin  d  -f-  cos  n  cos  d  cos  (t  —  I  —  m).    (358) 

Such  an  equation  is  given  by  each  thread  observed.  If  ^ 
threads  are  observed,  the  mean  of  the  resulting  equations 
will  be 

c  +  z'0  =  —  sin  n  sin  d  -\-  cos  n  cos  d  —  2  cos  (/  —  m),    (359) 

where  t0  is  the  mean  of  the  equatorial  intervals,  2  is  the  sum- 
mation sign,  /  represents  the  hour-angle  corresponding  to 
any  thread. 


§  207-  VESSEL'S  METHOD   OF  REDUCTION.  361 

Let  T  =  the  arithmetical  mean  of  the  times  observed  on 
the  individual  threads  (supposed  corrected  for 
clock  error  and  rate) ; 

T  —  I  =  the  time  over  any  thread. 

Then  (t  —  m)  =  (T  —  a  —  m)  —  I, 

and 

-2  cos  (t  —  m)  =  cos  (T—  a  —  m)  -2  cos  / 

-f-  sin  (  T  —  a  —  m)  —2  sin  /. .     (360) 

Now  let  k  cos  K  =  —  2  cos  /; 

k  sin  K  =  —  2  sin  I. 


Then     —  -2  cos  (/  —  m)  =  k  cos  (T  —  a  —  K  —  m).      (362) 

(359)  then  becomes 

c  -\-  t0=  —  sin  n  sin  tf  +  k  cos  ^  cos  tf  cos  (T  —  a  —  u  —  m). 

Now  let  y  cos  dl=.  k  cos  d  ; 

y  sin  ^  =      sin  6. 


Then  (363)  becomes 

c  -\-  i 

-  =  —  sin  n  sin  tfj-j-  cos  #  cos  tf,cos  (T—  a—  H  — 

Thus,  by  computing-  the  auxiliary  quantities  y,  ^,  and  K, 
the  form  of  the  equation  for  the  mean  of  the  threads  is  the 
same  as  that  for  the  middle  thread. 

Practically  y  will  seldom  differ  appreciably  from  unity. 


362  PRACTICAL  ASTRONOMY.  §  2O/. 

£,  and  H  may  very  readily  be  computed  by  the  aid  of  tables 
A  and  B,  page  365.  These  tables  are  computed  as  follows: 
Since  21  =  o  (T  being  the  mean  of  the  observed  times, 
.and  /the  difference  between  Tand  the  time  on  any  thread), 
(361)  may  be  written 


k  cos  H  =  I  —  —  2  sin3  £/; 
k  sin  H  =      —  —  2(1  —  sin  /). 


(366) 


From  these  it  appears  that  k  sin  n  is  of  the  order  73,  and 
that  k  cos  H  only  differs  from  unity  by  a  quantity  of  the  order 
7a.  There  will  then  be  no  appreciable  error  in  writing 


*=       -  -2(I-  sin/). 
And  since,  from  (364),  we  have 

tan  dl  =  \  tan  8,  ......     (368) 

K 

the   method   of  Art.  74  for  expanding  a  function  of  this 

form  gives 

-  k  sin  2<?    .    I    i  -  k    sin  4^ 

•  (369) 


This  becomes,  by  substituting  for  k  its  value, 

l5S5Li£ 

^=d  *      Sinl//-sin2^  .....     (370) 


Sill 


For  computing  ^,  table  A,  page  365,  gives  the  value  of 
^;  the  argument  being  th'e  difference  between  each  ob- 

X         * 


§  207-  BESSEL'S  METHOD   OF  REDUCTION.  363 

served  time  respectively  and  the  mean  of  all,  expressed  in 
minutes  and  seconds  of  time  for  convenience.  The  arith- 
metical mean  of  these  quantities  will  be  the  numerator  of  the 
coefficient  of  sin  28  in  (370).  The  denominator  differs  very 
little  from  unity.  When  desirable,  this  small  difference  may 
be  corrected  by  table  B,  the  argument  of  which  is  the  numer- 

i       sin2^/ 

ator,  viz.,  -  2  -.  -  /7. 
'  ^      sin  I 

The  fourth  column  of  table  A  gives  the  quantity  (/  —  sin  /), 
the  arithmetical  mean  o{  these  quantities  being  equal  to  n. 
If  y  is  required,  we  readily  find,  from  (364), 

_  i  —  (i  —  k)  cos2# 
cos(^-  tf)~~' 

The  denominator  does  not  differ  appreciably  from  unity,  and 


Therefore  y  =  i  —  -  cos'tf  2  sin8  £/.    .    .    .    .    (371) 

Since  this  only  appears  as  the  divisor  of  the  small  quan- 
tity c  +  z'0,  it  will  very  rarely  be  required. 

The  quantity  *0  will  vanish  when  the  star  is  observed  over 
all  of  the  threads,  and  the  equatorial  intervals  reckoned  from 
the  mean  of  the  threads. 

Having-  shown  how  our  fundamental  equation  which  ap- 
plies to  the  time  over  the  middle  thread  may  be  reduced  to 
a  like  form  when  the  time  is  the  mean  of  the  times  over  the 
different  threads  —  see  equation  (365)  —  we  may  now  solve  this 
equation  for  (p  as  before. 

Formulae  (349)  and  (350)  will  then  have  the  form 

tan  (p'  •=.  tan  d1  sec  (T  —  a  —  «)  cos  m^  \ 


PRACTICAL   ASTRONOMY. 


208. 


208.  Formula  for  Latitude  by  Prime  Vertical  Transits. 

Preliminary  Computation. 


sm 

COS  Z  =    - 


COS  t  = 


sn  q? 
tan  d 
tan  c' 


7  =  ^--    ^— ; 

sm  cp  sin  £ 

Clock  time  of  passing  first  thread 


Reduction  to  Middle  or  Mean  Thread. 
j _* 

sin  ((p  —  b)  cos  d  sin  ($  — 


w  =  ^  [7" 
tan   >'  =  tan  (J  sec  S  cos 


T+ 


Bessel's  Method  of  Reduction. 

~  -        I  ~  sin  /; 


sn 
sin  \ 


sm 


tan  ^r  =  tan  d,  sec  (  7"  —  <*  —  K)  cos  m\ 


(XX) 


(XXa) 


(XXb) 


209. 


EXAMPLE  OF  PRIME  VERTICAL  TRANSITS. 


365 


TABLE  A. 

For   reducing   transits   over  several  threads  to  a 
common  instant. 


sin2  £7 

D 

7 

sin2  £7 

D 

sin  i" 

sin  i" 

om  oo8 

o".oo 

".00 

6moo8 

35".  34 

1.94 

".62 

IO 

20 

30 
40 

50 

•03 
.11 
•25 
•44 
o  .68 

!o8 
.14 
.19 
•24 
•3° 

.00 

.00 

.00 
.00 
.00 

IO 
20 

3° 
4° 
50 

37  -33 
39  -38 
41  .48 
43  -63 
45  -84 

1.99 
2  05 

2  .  IO 

2-15 
2.21 
2.26 

.67 

•73 
•79 
•85 
.91 

ImOO« 
10 

o  .98 
1  -34 

•  36 

.00 

.00 

7m  oos 

IO 

48  .10 
50  .42 

2.32 
2  .  37 

.98 
•05 

20 

i  -75 

46 

.01 

20 

52  -79 

2  43 

.  12 

3° 
40 

2  .21 

2  -73 

•52 

•  57 

.01 

.01 

30 
40 

55  -22 
57  -70 

2  48 

2  •  54 

.20 

.28 

So 

3  -3° 

•63 

.02 

50 

60  .24 

2.59 

•37 

2mOO" 
IO 
20 

3  -93 
4  .61 

5  -35 

.68 
•74 

.02 

.04 

8moo« 

IO 
20 

62  .83 

65  -47 
68  .17 

2.64 
2.70 

.46 
•55 
•65 

3° 

6  Vi4 

•79 
.84 

.04 

3° 

70  .92 

2.75 

2.81 

•75 

5° 

6  .98 
7  .88 

.90 
.96 

:°ol 

40 

50 

73  -73 
76  -59 

2.86 
2.92 

.86 
•97 

3moo8 

IO 
20 

8  .84 
9  -84 
10  .91 

.00 
.07 

.  12 

.08 
.09 
.11 

9m  oo8 

10 

20 

79  -51 
82  .48 
85  -5! 

2  97 
3-°3 
3.08 

.08 

.20 

•32 

3° 

40 

12  .03 
13  .20 

•17 

.12 
.14 

3° 
40 

88  .59 
91  -73 

3-i4 

'si 

5° 

14  -43 

!z8 

.16 

50 

94  -92 

3-  T9 
3-24 

•  72 

4ra  oo8 

IO 

20 

3° 
40 

5° 

15  -71 
17  .04 
18  .43 
19  ,88 

21   .38 
22  .93 

•33 
•39 
•45 
•So 
•55 
.61 

.18 
.21 

:3 

.29 
•32 

iom  oo8 

10 

20 

30 
40 
50 

98  .16 
101  .46 
104  .81 

I08  .22 

ii  .68 

15  .20 

3-30 
3-35 

3-52 
3-57 

2.86 
3-oo 

3-30 
3-46 
3-63 

10 

20 
3° 

24  -54 
26  .21 
27  .92 
29  .70 

.67 
•7i 

•36 
.40 
•44 
.48 

IImOO" 
IO 

20 

30 

18  .77 

22  .39 
26  .07 
129  .8l 

3.62 
3-68 
3-74 

3-8o 
3-98 
4.16 
4-34 

4° 
50 

31  .52 
33  -40 

'.S3 

•52 
•57 

4° 
50 

133  -60 

137  -44 

3  •  79 

3-84 

4-53 
4-73 

TABLE  B. 

For  correcting  the 
coefficient  of  sin  26. 


i  2  sin**/ 
fj.   sin  i" 

Correc- 
tion. 

| 

10" 

+  ".000 

20 

.002 

3° 

.004 

40 

.008 

5° 

.012 

60 

.017 

70 

.024 

80 

.031 

90 

•039 

IOO 

.048 

no 

•059 

120 

.070 

I30 

.082 

I40 

•095 

ISO 

.109 

160 

.124 

170 

.140 

1  80 

•157 

190 

•I75 

200 

.194 

209.  As  an  example  of  the  determination  of  latitude  by  this  method,  the  fol- 
lowing observations  have  been  selected  from  Pierce's  Memoir  on  the  Latitude 
of  Cambridge,  Mass.  (Memoirs  of  American  Academy  of  Sciences,  vol.  ii.  p.  183); 


366 


PR  A  CTICAL  A  S  TRONOM  Y. 


§209. 


STAR. 

Date. 

1884. 

d 

6 

M 

u 

Transit. 

TIMES  OF  TRANSIT  OVKR  THREADS. 

Error  of 
Level. 

N.  end 
high. 

<l 

<b 

*3 

'4 

'6 

*« 

'7 

a  Lyrae 

Dec.  23 

S. 

S. 

E. 
W. 

I6hs8m     58.8 
20    21     45  .0 

36m57'.2 
22    54  .0 

35m  49"-o 
24      i  -5 

34m  428.o 
25      9-o 

33m34"-5 
26    16  .2 

32m  288.2 
27     22  .5 

3im22".5 
28    28  .1 

+    ".41 

—           .02 

a  Lyrae 

Dec.  29 

N. 
N. 

E. 
W. 

l6     31       22  .3 
20     28      28  .3 

32  28  .5 

27      22  .5 

33    34-3 
26    18  .4 

34    4i  -2 

35    47  -1 
24      2  .0 

36    55  -2 

37      5  -5 

-  i".25 

-    I     .32 

|3  Persei 

Dec.  25 

N. 
N. 

E. 
W. 

I      26      29  .O 

4    26     17  .8 

27    57  .0 
24    49  -5 

29    25  .0 

23      21   .4 

30    55  -5 
21    50  .8 

32    27.8 

20      19  .0 

34      1-5 
18    45  -5 

35    36-5 
17    10  .0 

±           >IS 
-f          .22 

ft  Persei 

Dec.  26 

S. 
S. 

E. 
W. 

i    35     36  -5 
4    17     ii  -o 

34      0.5 
1  8    46.0 

32      27  .5 

20    19  .6 

30    55  -6 

21      51   .O 

29     24  .5 

27    56.5 

26    28.5 

+          .87 
+          -87 

The  equatorial  intervals  of  the  threads  from  the  middle  thread  are 

*i  —  5iMi;          i*  =  33s.  98;          **8  =  1 7s.  02;          ^^  =  o8.oo;          i&  =  I78.io; 
it  =  34s.  14;  i-,  =  5IM6. 

The  clock  correction  and  rate: 


Date. 

Sidereal 
Time. 

&T. 
Clock  slow. 

Daily 
Rate. 

Dec.  20 

24 

oh  go™ 
2        0 

+  Im  46s.  83 
-fl     48.74 

-       '.46 
—         81 

25 
26 

2      15 

2     45 

+  i     49-55 
+  i     47  .78 

+    L77 

Jan.    2 

o    30 
5     15 

+  i     48.13 
-j-  i     51-10 

-        .71 

Apparent  places  of  the  stars  observed : 

a  Lyrae,   December  23d,  a  —  i8h  3im  4O8.32;       S  =  38°  38'  39". 76. 

a  Lyrae,   December  2gth,  a  =  18    31    40.36;       6"  =  38    38  38  .08. 

ft  Persei,  December  25th,  d  =  40    21   25  .83. 

ft  Persei,  December  26th,  d  =  40    21   25  .86. 

The  collimation  error  c  is  assumed  equal  to  zero.    Assumed  <p  =  42°  22'  48". 
We  shall  first  compute  the  latitude  by  formulae  (XXa).     The  transits  over  the 
several  threads  must  first  be  reduced  to  the  middle  thread  by  the  formula 


7  = 


sin  (<p  —  b)  cos  8  sin  (5  —  i/)' 

The  complete  reduction  is  given  for  the  observations  of  a  Lyrae,  December  23d, 
in  order  to  illustrate  the  process. 


§  2O9-        EXAMPLE  OF  PRIME    VERTICAL    TRANSITS. 


367 


Observed 
Times. 

#. 

Observed  /. 

*/. 

*-*/. 

i6h  38™    5"-8 

-  3m  238.8 

-  25'  28" 

-  28°  22'  55" 

36    57-2 

—  2      15  .2 

-  16   54 

—  28  31  29 

35    49  -° 

-  i       7  .0 

-    8   23 

—  28    40     o 

34    42  .0 

33    34-5 

+  1    7.5 

-f    8    26 

—  28    56   49 

32    28  .2 
31    22.5 

3h  som  27".o 

+   2      13.8 

+  3     19.5 

+  16   43 

+  24   56 

—  29      5      6 
-  29    13    19 

i     55    13  -5 

20     21    45  .0 
22    54  .0 

28°  48'  23". 

-f  3    24  .0 

+2      15  -0 

+  25    30 
+  16   52 

+  28    22    53 
28    31    31 

24      i  .5 

-f-i       7-5 

-(-    8   26 

28    39   57 

25      9  .0 

26    16  .2 

-  i      7.2 

-     8    24 

28    56  47 

27     22  .5 

-  2      13  .5 

—  16  41 

29      5     4 

28     28  .1 

-3     19  -1 

-  24   53 

+  29    13    16 

sin  (#  -  i/). 

log 
Denominator. 

log/. 

/. 

Reduced  Time. 

9.67701 

9-39837 

2.31014 

—  204s.  2 

lfih  34m  4I9.6 

9.67901 

9.40037 

2.13085 

-  135  -2 

42  .0 

9.68098 

9.40234 

1.82862 

-  67  .4 

41  .6 

42  .0 

9.68485 

9.40621 

1.82679 

+  67.1 

41  .6 

9.68673 

9.40809 

2.12517 

+  133  -4 

41  .6 

9.68859 

9.40995 

2.29898 

+  199.1 

16    34    41  .6 

T  = 

16    34    41  .71 

9.67700 
9.67902 

9-39836 
9.40038 

2.31015 
2.13084 

-\-  204  .2 
+  135  -2 

20     25        9  .2 

9-2 

9.68097 

9.40233 

1.82863 

4-  67.4 

8.9 

9.0 

9.68484 

9  .  40620 

1.82680 

-  67.1 

9  .1 

9.68673 

9  .  40809 

2.12517 

-  133  -4 

9  .1 

9.68858 

9.40994 

2.29899 

-  199-1 

20    25      9  .0 

T'  = 

20    25      9  .07 

In  the  above  the  quantity  5  is  computed  from  the  second  of  (XXa)^  using  for 
T'  and  T  the  time  over  the  middle  thread,  and  neglecting  the  rate,  which  will 
be  less  than  the  probable  error  of  the  observation.  The  "observed  /"  is 
found  tiy  subtracting  the  observed  time  over  each  thread  from  the  time  over 
the  middle  thread.  The  quantities  headed  "  log  denominator"  are  computed 


368 


PRACTICAL   ASTRONOMY. 


§209. 


by  writing  the  quantity  log  (sin  (p  cos  8)  on  the  lower  edge  of  a  slip  of  paper 
and  adding  it  in  succession  to  each  of  the  quantities  in  the  previous  column,  b 
is  neglected  in  the  quantity  sin  (q>  —  b).  The  quantities  log  ii,  log  z'2,  etc.,  are 
then  written  in  order  on  the  lower  edge  of  another  slip  of  paper  and  the  "  log 
denominator"  subtracted,  giving  log  /.  It  would  be  sufficient  to  compute  the 
intervals  /  for  one  transit  only,  as  they  are  the  same  for  both;  but  in  a  case  like 
the  above  it  is  well  to  compute  both  as  a  check  on  the  work.  In  the  above,  four- 
figure  logarithms  would  have  been  sufficiently  accurate. 

In  the  same  manner  the  other  observations  are  reduced,  the  quantities  T  and 
T'  being  those  given  in  the  following  computation: 


Dec.  23. 


Dec.  29. 


Latitude  from  a  Lyra. 
CLAMP  SOUTH. 
T'  =  20"  25m    9s.  07 

AT' 

+ 

i 

48 

•73 

tan  ~d  —  9.9028502 

T'  +  AT' 

= 

20 

26 

57 

.80 

sec  3  =    .0573745 

A  T')  —  (  T  +  A  T} 

= 

3 

50 

27 

•53 

cos  ;//  =                oo 

3 

= 

i 

55 

13 

.765 

tan  <p'  —  9.9602247 

= 

28° 

48' 

26" 

•5 

qj  =  42°  22'  47" 

.68 

T 

= 

i6h 

34m 

4is-7i 

AT 

— 

-4- 

i 

48 

.56 

T+  AT 

= 

16 

36 

30 

.27 

AT-\-  T'  -j-  AT') 

= 

18 

31 

44 

•035 

a 

— 

18 

3i 

40 

•32 

m 

— 

+ 

3 

.715 

=, 

55 

"•7 

CLAMP  NORTH. 

T' 

= 

20h 

25m 

10s 

.12 

tan  d  =  9.9028429 

AT' 

= 

i 

48 

.72 

sec  3  =      0573924 

T'  +  AT' 

= 

20 

26 

58 

.84 

cos  m  =                 o 

AT1)  -  (T-}-  AT) 

= 

3 

50 

29 

•58 

tan  pi   =  9.9602353 

3 

= 

i 

55 

14 

•79 

= 

28° 

48' 

41" 

•9 

<pi'  =  42°  22'  50". 

19 

T 

= 

16" 

34m 

40" 

.66 

AT 

= 

4 

-  i 

48 

.60 

mean  <p'  =  42°  22'  48". 

935 

T  +  AT 

— 

16 

36 

29 

.26 

mean  b  =                —  . 

545 

\-  AT  +  T'  -{-  AT')  = 

18 

3i 

44 

•05 

a 

= 

18 

3i 

40 

•36 

<p  =  42°  22'  48". 

39 

m 

= 

+ 

3 

.69 

> 

— 

+ 

55" 

•3 

§210.       EXAMPLE  OF  PRIME    VERTICAL    TRANSITS.  369 

In  a  manner  precisely  similar,  from  the  observations  of  ft  Persei  on  Decem- 
ber 25th  and  26th  we  find — 

Dec.  25,  (f>   —  42°  22'  48".  50 

Dec.  26,  q>'  =  42    22   48  .56 

Mean        42    22  48  .53 

Mean  of  the  four  level-readings  -(-  -53 

<p  =  42    22  49  .06 

The  mean  of  these  two  determinations  from  a  Lyra  and  ft  Persei  is  therefore 

q>  —  42°  22'  48". 73. 

The  value  given  in  the  memoir  from  which  these  observations  are  taken   is 
42°  22'  48". 60.     This  is  the  result  of  a  long  series  of  observations. 


Application  of  Bessel's  Met  hod. 

210.  As  an  example  of  Bessel's  method  of  reduction,  let  us  apply  formulae 
(XXb)  to  the  foregoing  observations  of  a  Lyr<z. 


370 


PRACTICAL  ASTRONOMY. 


§210. 


0*        2   °   w  vo" 
tx         N          0V- 


INI  II  II  ON 


II      II  II  II 


s-'<«  G  II  e 

SJ83:o.         § 


II   II    II    II  o,  ||    ||    ||    || 

•0*7  8  ^  *  ^-7  S  :e- 


OO         VO          ONVO  vq"       "V  (S          ^t- 

%    £    II  II  II    I 


CO  "-. 

5?  2 


II        II      '« 

60     " 


si 


II     II     « * 


u  c3 


I    *«S 

-    5f5p 


-aooc^oo^o         co      V 
e  T 


"^H  00    O  OO  00    ^-  O 


I!  II  II  II  II      II      H      II 


II  II  II  II  II     II     II     II 

*    s    s 


.  M  VO     M  10 

• 


•^-  CO  tx  N    O    ""> 

q  t>.  t~.  co  q  -*oo 

00  00  VO    O         ^^j 

^    IO    ^-  M    ^^ 

e 

IO  M  VO     M  lO  OV 

"^000          M«> 

u  u  u  u  u  -H- 

u 


Vs.  as- 


II  II  II 
« 


II  II 


i  i 


S  ff°  °  °  q1! 
++  I 


^S    ON  N    8  "    O»  ^ 


ONO> 


vo   O 


VO  vo   O         M 


•*  o 


I  i  I++++         -H-+I  i  i  i 


1  1+  + 


1^1 
ft 


00    N  00          « 


N  00 


§  2IO.       EXAMPLE   OF  PRIME    VERTICAL    TRANSITS.  371 

In  this  computation  the  quantities  — — V  and  —  n  are  taken  from  table  A. 

From  the  values  of  the  equatorial  intervals  already  given,  we  find  for  the  ob- 
servations over  all  of  the  threads  —  z'0  =  ±  ".621.  The  west  transit  of 
December  2gth  being  observed  only  on  threads  I,  II,  III,  and  V,  we  have 
z'0=  —  318". 787.  c  is  assumed  equal  to  zero. 

The  correction  (c  -\-  *0)  -: — ~-  is  appreciably  the  same  for  the  two  transits  of 

December  23d  and  for  the  east  transit  of  December  291*1,  viz.,    ±  ".67.     For 
the  west  transit  of  December  2gth  the  computation  of  this  term  is  as  follows: 

log  (f+io)  =  2.5035oo6« 

sin  (fl  =  9.8295232 
cosec  6\  =    .2044758 
log  correction  =  2.5374996,* 
correction  =  —  344".  746 

Then  we  have,  December  23d, 

E.  tp  =  42°  22'  33".  12        W.  (f>  =  42°  23'    3".64 
b  —  +  .41  —  .02 

«+'«>^f  = 

cp  •=.  42°  22'  32". 86  42°  23'    2".g5 

Mean  q>,  Dec.  23d,  clamp  south,         42°  22'  47". 90 

Dec.  2gth,  E.  <p'  =  42°  22'  34". 40  W.  <p'  =  42°  28'  50". 35 

•    b  =            —  i  .25  —         i  .32 
.  .  sin  <p' 

('  +  'o)-shTs;=  -544-75 

<p  =  42°  22'  33". 82  42°  23'    4". 28 

Mean     ,  Dec.  2gth,  clamp  north,         42°  22'  49". 05 
The  mean  of  the  two  values  is  tp  =  42°  22'  48". 47 

It  will  be  observed  that  the  corrections  given  in  table  B  are  here  inappreci- 
able, y,  computed  from  formula  (371)  for  the  west  observation  of  December 
2gth,  is  found  to  be  0.99998433;  dividing  the  quantity  (c  +  »0)  by  this  factor 
(365).  we  find  for  the  correction  344". 752,  instead  of  344". 746  found  by  neglect- 
ing this  factor.  The  difference  is  inappreciable  in  this  case. 


3/2  PRACTICAL   ASTRONOMY.  §211. 


Application  of  the  Method  of  Least  Squares  to  Prime  Vertical 

Transits. 

211.  In  the  preceding  discussion  we  have  supposed  the 
stars  observed  at  both  the  east  and  west  transits,  and  in  both 
positions  of  the  axis.  The  method  is  very  simple  theoreti- 
cally, and  the  results  very  satisfactory.  In  the  tield,  time 
will  sometimes  be  wanting  for  applying  it  in  the  manner 
there  explained.  Besides  this,  many  observations  would  or- 
dinarily be  lost  by  the  interference  of  clouds  at  the  time  of 
one  transit  or  the  other.  For  meeting  these  difficulties  the 
following  modification  will  be  useful : 

A  number  of  stars  must  be  observed,  some  east  and  some 
west,  the  axis  being  reversed  about  the  middle  of  the  series. 
Care  must  be  taken  to  observe  about  an  equal  number  in 
both  positions  of  the  axis,  and  about  the  same  number  of 
east  and  west  stars.  The  declinations  of  stars  observed  east 
should  be  as  nearly  as  may  be  the  same  as  those  observed 
west. 

We  shall  suppose  the  observations  reduced  to  the  middle 
or  mean  thread  by  the  method  of  Bessel  (Art.  207);  then  in 

?+  i 
equation  (365)  let  us  write  r^=  T—a—nand —c'.  Then 

expanding  cos  (r  —  m),  the  equation  becomes 

cf  =  —  sin  n  sin  tfa  +  cos  n  cos  m  cos  ^  cos  rl 

-\-  cos  n  sin  m  cos  tf,  sin  r,.  .     (373) 

Now  substituting  for  sin  n,  cos  n  cos  ;«,  and  cos  >n  sin  m, 
their  values  from  (342),  this  becomes 

cf  —  —  cos  (cp  —  b)  sin  dl  -j-  sin  (cp  —  b)  cos  dl  cos  rl 

+  a  cos  dl  sin  rr     .     .     .     (374) 


§211.  REDUCTION  BY  LEAST  SQUARES.  373 

Let  the  auxiliaries  <p,  and  z  be  determined  by  the  equa- 
tions 

cos  z  sin  cpl  =  sin  <S\;  \ 

COS  ^  cos  q>l  —  cos  6l  cos  r,;  v  .    .     .     .     (375) 
sin  z  =  -cos  tf,  sin  r,.  ) 

Then  (374)  becomes 

c'  =  sin  (cp  —  cp^  —  b)  cos  ,0  -|-  #  sin  z. 

Since  sin  (cp  —  cp1  —  b)  is  here  of  the  same  order  as  a  and 
c' ,  we  may  write  this  equation 

cp  —  cpl—  b-{-a  tan  s  —  c'  sec  s  ==  o.      .     .     (376) 

Now  let  (p  =  cp0  -\-  dtp,  in  which  q>0  is  an  assumed  approxi- 
mate value  of  (p.  Then  writing  f=  <pQ—  <pl  —  b,  viz.,  the  al- 
gebraic sum  of  the  known  terms,  we  have 

Acp  -\-a  tan^  —  c  sec^r  -|_y—  o.     .     .     .     (377) 

Each  star  observed  furnishes  one  equation  of  this  form 
for  determining-  the  unknown  quantities  Acp,  a,  and  c.  A 
considerable  number  of  stars  should  be  observed,  and  the  re- 
sulting equations  solved  by  the  method  of  least  squares. 

The  formulae  for  this  method  are  then  as  follows  • . 

rl=  T  —  a  —  H  ; 

cos  z  sin  cpl  =  sin  (^ ; 

cos  z  cos  <pt  =  cos  tfj  cos  TI  ; 


sn  s  —  cos  <J1  sin 

-)-  a  tan  z  —  c'  sec  # 


(XXI) 


9  =  <Po 
H  and  #j  are  determined  as  explained  in  Art.  207. 


374 


PRACTICAL  ASTRONOMY. 


§211. 


Example. 

The  following  observations  were  made  at  Munich  by  Bessel,  1827,  June  28th, 
with  a  small  transit  instrument  mounted  on  a  tripod  and  approximately  adjusted 
in  the  prime  vertical:* 


<J 

TIMES  OF 

TRANSIT  OVER 

THREADS 

STAR. 

£ 
2 
U 

i 

H 

<j 

'2 

*1 

>4 

*5 

Level. 

A  Bootis  
a  Lyrse  

S. 

s 

W. 
F 

i8m    o'.o 
28    32  .8 

i5m  52".4 
29    20  .24 

I0h  I3m  42s  8 

30      8  .8 

IIm  23».2 

30    58  .8 

8m  so'.o 
31     50  .8 

—    2d.II3 

XIII  316 

N 

W 

4.Q         4    8 

i  Herculis  

N. 

N 

E. 
E 

8     38  .0 

6    50.8 
Azimuth 

ii      5      5-6 
disturbed. 

3      21  .2 

i     32  .0 

-    .798 

v  Herculis  
y  Cygni  

N. 
N 

W. 

F 

5     53  -2 
25     58  .0 

7     54  -o 
25       5  .6 

12      9    52  .8 

24       13  .2 

II       52  .0 

23     23  .6 

13  52  .4 

—         .  122 
—         .123 

<f>  Herculis  
8  Cvffni 

S. 

s 

W. 
F 

41       7  .2 

39     4°  -4 

12      38       II   .2 

36     38-8 

35      o.o 
48     14   4 

—       -353 

Th<-  -,•  ^r  nt  places  of  the  stars  for  the  date  of  observation,  1827,  June  28th, 
i6b  34ni,  Munich  sidereal  time,  I  find  to  be  as  follows: 


STAR. 

a 

£ 

A  Bootis   

I4h     Qm  SO8  2O 

46°  t;V  i^//.4o 

18    31       8  .14 

38    37    40    .OI 

XIII  316 

14.       I         2    ^6 

44    4O    ^    .^^ 

i  Herculis             .  . 

17    7.1      ^8  .04 

46       6    2O    .  56 

18    50       7  .75 

43    43   28    .14 

v  Herculis 

ic     c7      27    4s; 

46    ^i    2^    .^o 

Y  Cvcrni 

20   16      4   61 

on     42     74    .46 

cp  Herculis  

16     3     21  .83 

45    23   40   .34 

iq    -JQ     38  .03 

44   42    52   .86 

The»values  of  the  equatorial  intervals  of  the  threads  from  the  mean  thread 
are  as  follows  : 


.  19  ;   *4=  — 


*6=—  612".  46. 


The  correction  for  inequality  of  pivots  is  —  0.294!  divisions  of  level  for 
circle  north.     The  value  of  one  division  of  the  level  is  4".  49. 


*  See  Astronontiscke  Nachrichten,  vol.  ix.  p.  415 


*  See  Astronomische  NacnrtcMen,  vol.  ix.  p.  415. 

t  Bessel  uses  as  the  correction  —.42  divisions,  which  is  evidently  computed  by  the  erroneous 

>  ^  -  A  instead  of  (297).    See  Ast.  Nach.,  vi.  p.  236. 

-f~  COS  *!/ 


formula  /  =  B>  ~  B  ( 

2  NCOS 


§  211.  EXAMPLE   OF  REDUCTION  BY  LEAST  SQUARES.        375 

A  mean  time  chronometer  was  used,  the  hourly  rate  on  sidereal  time  being 
-j-  9s.  19  ;  the  correction  at  12  hours  chronometer  time  being  5h  4™  44". 61. 

Bessel  gives  the  approximate  values  of  the  latitude  and  the  azimuth  of  the 
instrument  as  follows  : 

<p0  =  48°  8'  40"  ; 
aQ  =    o°  7'  48". 

If  these  quantities  are  not  known  with  accuracy  sufficient  for  forming  the 
equations  of  condition,  a  preliminary  reduction  of  a  few  of  the  observations  will 
give  them. 

The  values  of  T,  H,  and  di  are  computed  precisely  as  shown  in  Art.  210.  With 
this  series  of  observations  u  in  no  case  exceeds  §.oi  ;  it  has  accordingly  been 
neglected. 

The  computation  of  TI  for  each  star  may  now  be  conveniently  arranged  as 
follows: 


STAR. 

T 

AT 

T+AT 

a 

'. 

Tj 

. 

a  Lyrae  

E. 

10  30    10  .29 

4    3°  -85 

15  34    41  .14  18  31      8  .14 

-f-  ih  8m  n*.79 
—  2  56    27  .00 

-f-i7     2  50   .85 
—44     6  45   .0 

XIII  316... 
/'  Herculis. 

W. 
E. 

10  47    44  .32 
"     5      5-52 

4    33  -55 
4    36  .20 

15  52    17  .87  14     i      2  .36 
16    9    41  .72  17  34    38  .04 

+  i  Si    15.51 
—  i  24    56  .32 

+27   4852   .65 
—21    14    4   .8 

77  l.yrse.   .  . 

E. 

ii  41    38  .90 

4    41  .80 

16  46    20  .70  18  50      7  .75 

-  2     3    47  .05 

—3°  56  45   .75 

71  Herculis. 

W. 

12     9    52  .88 

4    46.13 

17  14    39  .01  15  57    27  .45 

+  i  17    ii  .56 

+19    17  53   .4 

Y  Cygni  .    . 
$  Herculis. 

E 
W. 

12    24     40  .IO 

12  38      7  .52 

4    48.39 
4    50.45 

17  29    28  .49  20  16      4  .61 
17  42    57  .97;  16    3    21  .83 

—    2    46     36  .12 

+  i  39    36  .14 

-41    39    i    -8 
+24   54    2   .i 

SCygni.... 

E. 

12  45    26  .32 

5  4    51  -57 

17  50    17  .89  19  39    38  .03 

—   i  49    20  .14 

—  27    20     2     .1 

I 

As  we  have  an  approximate  value  of  the  azimuth  error,  we  may  write  (equa- 
tion 376) 

cp0  -\-  A(p  —  <P!  —  b  -}-  (a0  +  do)  tan  z  —  (t0  -f-  0  sec  *  =  o. 

to  is  zero  for  all  the  above  stars  except  ft  Lyra  and  y  Cygni.  In  the  observa- 
tion of  Tt  Lyra  the  transit  over  the  second  thread  was  lost.  Therefore  for  this 
star  z0  is  the  mean  of  the  equatorial  intervals  *i,  z'3,  i±  «6;  viz.,  —  75".  775- 

Similarly  for  y  Cygni,  the  fifth  thread  being  missed,  *0  =  +  153".  1  125. 

Writing  the  sum  of  the  known  terms,  viz., 


<po  —  \_<p\  +  b  —  a0  tan  z  -f-  *„  sec  z\  = 
our  equation  of  condition  becomes 

Aq>  -j-  da  tan  z  —  c  sec  z  =.  —  f. 


3/6  PRACTICAL  ASTRONOMY.  §211, 

The  computation  of  q>i,  tan  z,  sec  z,  and  f  is  now  arranged  as  follows  : 


A  Bootis. 

a  Lyrae. 

XIII  316. 

i  Herculis. 

n  Lyrae. 

«, 

46°  53'  25"-68 

38°  37'  5o"-33 

44°  40'  57".22 

46°    6'26".74 

43°  43'  3i"-25 

tarn  &! 
cos  TJ 
tan  <f>j 

.0286798 
9.9804823 
.0481975 
48°  10'  22".  10 

9.9026368 
9.8561090 
.0465278 
48°    3'47"-93 

9.9951876 
9.9466792 
'      .0485084 
48°  n'  35".47 

.0167925 
9.9694648 

48°    6'  5^//B7^ 

9.9806703 
9-9333"i 
.0473592 
48°    7'    4".22 

tan  T 

log  tan  2 
log  sec  2 

9.48667 
9.82405 
9.31072 
.00890 

9-98655n 
9  82498 
9-8"53n 
.07611 

9.72228 
9.82388 
9.54616 
.02532 

9-82454 

9.4i4oin 

.014I4 

9-77784n 
9.82452 
9.602360 
.03228 

tan  2 
sec  2 

4  -2045 

1.0207 

—    .6479 
1.1915 

4-  -3517 

i.  0600 

-  .2594 
1.0331 

-     -4003 
1.0771 

Level-reading  
Inequality  of  pivots 
b 
t'o  sec  z 

—  a0  tan  2 

-2.113 

4  -294 

8".i7 

-    i'35".7' 

-  2.340 

4  -294 

4-     5'    3"-24 

-  1.132 
—    .294 

-      2'  44"-59 

-  1.798  • 
-  .294 

-             9"-39 

4    -105 
-    -294 
o".85 

—         i'  2l".62 

4      3'    7"-33 

[_           -j-  *0  sec  2  J 
f   = 

48°      8'38".22 

48°    8'  41".  98 
i".98 

48°    8'44//.48 
4".48 

48°    8'  48".  80 
8".  80 

48°    8'49".o8 
9".o8 

v  Herculis. 

y  Cygni. 

<J>  Herculis. 

8  Cygni. 

», 

46°  31'  3i".34 

39°  42'  35"-37 

45°  23'  44"-94 

44"  42'  s6".6o 

tan  «! 
COSTX 

tan  </>! 

.0231351 
9.9748853 
.0482498 
48°  10'  34".  43 

9.9193426 
9.8734442 
.0458984 
48°    i'  ig".32 

.0060007 
9.9576263 
.0483744 
48°  xx'    3".84 

9.9956904 
9.9485819 
.0471085 
48°    6'    s//.o4 

tan  T 

log  tan  2 
log  sec  z 

9-54427 
9.82402 
9.36829 
.01153 

9.949110 
9  82532 
9-77443n 
.06579 

9.66670 
9.82395 
9.49065 
.01986 

9-7I34°n 
9.82466 
9-538o6,, 
•02445 

tan  2 
sec  2 

4  -2335 
1.0269 

—   1.  122 
—      .294 

6".36 

-          i'  49".28 

48°    8'  38".79 

-    -5949 
1.1636 

-  2.123 
-    -294 
—           io".8<; 
4-      2'  58^.16 

48°    8'  45"  .04 
-             5"-04 

4  -3095 
1.0468 

4  '.'^ 
-             4"-75 

-       2'  24//.84 
48°    8'  34"-25 
4             5"-75 

—    -3452 
-1.124 

4  .294 

-              3"-73 
+       2'  41".  55 
48°    8'  42".86 
-             2".  86 

Inequality  of  pivots  

?'0  sec  2 
—  a0  tan  2 

t<4,  +  b  —  a0  tan  z~l 
+  i.  sec  2  J 

/    = 

Since  the  azimuth  of  the  instrument  was  disturbed  between  the  observation 
of  i  Herculis  and  it  Lyrae,  it  will  be  necessary  to  introduce  into  the  equations  a 


§  211.  EXAMPLE   OF  REDUCTION  BY  LEAST  SQUARES.        377 

different  value  of  the  azimuth  correction  for  these  stars  observed  after  the  dis- 
turbance took  place. 

The  equations  will  therefore  be  * 

c  —  o 

A  Bootis,     Acp-^-  .2045^0  —  1.0207*:  =  —  i".78  —    .50. 

a  Lyrse,     Acp  —  .6479^/0  —  1.1915*:  =  -f~  i^.gS  —    .08. 

XIII  316,     2/9>+  .3517^0  +  1.0600*-  =  -f-  4".  48  —2.35. 

i  Herculis,     Acp  —  .2594^0  +  1.0331^  =  -f-  8".  80  —  3.44. 

it  Lyrae,     Acp  —  .4003/^0'  -f-  1.0771*-  =  +'9//.o8  —  1.78. 

v  Herculis,     Acp  -\-  .2335^/0'  ~|~  1.0269*-  =  —  i".2i  -|~  3-28- 

y  Cygni,     Acp  —  .59494  a'  +  1.1636*-  =  -f-  5//-O4  -j-  4.05. 

cp  Herculis,     Acp  -\-  .3095/70'  —  1.0468*-  =  —  5".  75  -}-  2.01. 

d  Cygni,     Acp  —  .3452^0'  —  1.0579*:  =  -\-  2".  86  —  1.33. 

From  these  nine  equations  of   condition  the  following  normal  equations  are 
formed  : 


9.ooooAcp  —  .3511^/0  —  .  7974^0'  -{-    1.0438*:  =     23.5000;. 

—  .  351  1  Acp-{-  .6526^/0  '-f-      .668i<r  =  —  2.3539; 

—  .7974Z/^>  -f-  .7836^/0'  —       .8424*-  =  —  9.6825; 
i.  0438^/9?  -j-  .6681^/0  —  .8424^0'  -{-  10.4360*-  =      30.6933. 

Solving  these  equations  by  the  usual  methods,  we  find  the  following  values  : 

Acp  =  +  i".38  ; 
Aa  =  -  5".4i  5 
Ad  =  -  8".o7  ; 


Therefore  the  latitude  as  given  by  this  series  of  transits  is 

<p  —  48°  08'  41".  38. 

Bessel  gives  as  the  true  value  of  cp  found  from  other  sources  48°  8;  39".  50, 
from  which  the  above  value  would  be  only  i".88  in  error,  an  agreement  which 
is  very  satisfactory  when  it  is  remembered  that  the  instrument  used  was  a  very 
small  one,  mounted  quite  imperfectly,  and  used  in  the  open  air.  The  residuals 
given  in  connection  with  the  equations  of  condition  result  from  the  above 
values.  The  weights  and  probable  errors  may  be  computed  from  these  in  the 
usual  manner  if  thought  desirable. 

*  These  equations  are  not  the  same  as  those  given  by  Bessel  for  these  observations,  the 
differences  being  due  to  the  erroneous  value  of  the  correction  for  inequality  of  pivots,  before 
referred  to,  and  to  slightly  different  values  of  a  and  S  for  some  of  the  stars. 


CHAPTER  VII. 

DETERMINATION  OF  LONGITUDE. 

212.  The  difference  in  longitude  of  two  points  on  the 
earth's  surface  is  equal  to  the  angle  at  the  pole  formed  by 
the  meridian  curves  passing  through  the  two  points.  As 
the  earth  revolves  uniformly  on  its  axis,  it  will  be  equal  to 
the  difference  between  the  times  of  transit  of  the  same  star 
over  the  two  meridians,  and  may  be  expressed  either  in  de- 
grees, minutes,  and  seconds  of  arc,  or  in  hours,  minutes,  and 
seconds  of  time ;  for  astronomical  purposes  the  latter  desig- 
nation is  generally  preferred. 

Any  meridian  may  be  assumed  as  the  prime  meridian  from 
which  to  reckon  longitudes.  At  the  meridian  conference 
which  assembled  in  Washington,  October  1884,  Greenwich 
was  chosen  as  the  universal  prime  meridian.  Heretofore 
most  of  the  leading  nations  of  the  world  have  reckoned  lon- 
gitude from  the  meridian  of  their  own  capital.  In  conformity 
with  this  custom,  longitudes  within  the  limits  of  the  United 
States  have  been  reckoned  from  the  meridian  passing  through 
the  centre  of  the  dome  of  the  U.  S.  Naval  Observatory  at 
Washington.  For  local  purposes  the  meridian  of  Washing- 
ton will  no  doubt  continue  to  be  employed,  but  for  general 
scientific  purposes  longitudes  in  this  country  will  hereafter 
be  reckoned  from  Greenwich. 

As  an  astronomical  problem,  the  determination  of  the  dif- 
ference of  longitude  between  two  places  consists  in  an  ac- 
curate determination  of  the  local  time  at  each  place  and  the 


§213.  LONGITUDE  BY  CHRONOMETERS.  379 

comparison  of  the  times  so  determined ;  the  difference  be- 
tween the  times  being  the  difference  of  longitude. 

The  local  time  will  generally  be  determined  with  the  tran- 
sit; and  when  great  accuracy  is  required  in  the  resulting  lon- 
gitude, all  of  the  refinements  and  precautions  to  which  at- 
tention has  been  called  in  treating  of  this  subject  must  be 
observed.  For  rough  determinations,  especially  at  sea,  the 
time  is  determined  with  the  sextant  or  any  suitable  instru- 
ment. Nothing  need  be  added  on  this  point  to  what  has 
been  already  said.  We  shall  therefore  in  this  chapter  con- 
fine our  attention  to  the  practical  methods  of  comparing 
the  local  time. 

There  are  various  methods  which  may  be  employed  for 
comparing  the  local  time  at  two  meridians,  some  of  these 
admitting  of  a  much  higher  degree  of  accuracy  than  others. 
The  most  important  are  the  following: 

First.     By  transportation  of  chronometers ; 
Second.     By  the  electric  telegraph  ; 

Third.  Methods  depending  on  the  motion  of  the  moon, 
such  as  by  occultations  of  stars,  eclipses  of  the 
sun,  lunar  culminations,  and  lunar  distances. 

Also,  some  use  has  been  made  of  terrestrial  signals,  eclipses 
of  Jupiter's  satellites,  and  eclipses  of  the  moon. 

The  most  accurate  of  all  these  methods,  when  it  can  be 
employed,  is  the  telegraphic. 

Longitude  Determined  by  Transportation  of  Chronometers. 

213.  We  shall  designate  the  two  stations  whose  difference 
of  longitude  is  to  be  determined  by  E  and  W,  E  being  east 
of  W.  Let  the  error  and  rate  of  the  chronometer  be  deter- 
mined at  E  by  any  of  the  methods  given  for  determination 
of  time ;  then  let  the  chronometer  be  carried  to  W  and  its 


380  PRACTICAL  ASTRONOMY.  §213. 

error  on  local  time  determined  at  this  place^  The  difference 
between  the  time  at  W  given  by  observation  and  the  time 
at  E  which  will  be  given  by  the  chronometer  is  the  differ- 
ence of  longitude.  The  chronometer  may  be  regulated  to 
either  mean  or  sidereal  time.  To  express  the  difference  of 
longitude  algebraically, 

Let  A  T0  =  chronometer  correction  at  E  at  chronometer 

time  T0-> 

St  =  rate  per  day  as  shown  by  chronometer  ;* 
A  Tw  =  chronometer  correction  on  local  time  at  W  at 

chronometer  time  Tw\ 
A  =  difference  of  longitude. 

Then  (Tw  -f-  4TW)  =  true  time  at  W  at  chronome- 

ter time  Tw\ 
Tw  -f  A  T0  -f-  #*  (  Tw  —  TO)  =  the  corresponding  time  at  E. 

Therefore         1  =  A  Tw  -  (J  T0  +  dt  (Tw  -  T,)]    .     .     .     (378) 

Example.  At  Bethlehem,  Pa.,  1881,  August  7.75,  the  cor- 
rection to  a  mean  time  chronometer  was  found  to  be  -j-  6m 
50  '.90.  At  Wilkesbarre,  Pa.,  August  iod  9h  9m  i7s-92,  chro- 
nometer time,  the  correction  on  local  time  was  +  4'"  54s.  u. 
The  daily  rate  of  the  chronometer  was  +  is.64;  i.e.,  the  chro- 
nometer was  losing. 

Therefore  AT,  —  +  6m  5os-9O 

dt  —  '      is.6 


*2 

rw--r.)±=      2.  63  days 

dt(Tw-T9)  = 

4.31 

t 

Sum  .= 

6 

55.21 

^  = 

4 

54.11 

A     ITT 

2 

I  .1 

That 

is,  Wilkesbarre  is  2m  is.i 

west  of  Bethlehem. 

*  Unless  the  rate  is  uncommonly  large  it  will  make  no  difference  whether  we 
take  chronometer  days  or  true  days  in  applying  the  correction  for  rate. 


§214-  LONGITUDE  BY  CHRONOMETERS.  381 

214.  The  rate  is  determined  at  the  first  station  by  compar- 
ing the  results  of  observations  separated  by  an  interval  of 
several  days,  but  it  is  found  that  the  rate  of  the  chronometer 
during  transportation  (called  the  travelling  rate)  is  seldom 
the  same  as  its  rate  when  at  rest.  The  travelling  rate  may 
be  determined,  or  its  effect  may  be  eliminated  by  transport- 
ing the  chronometer  in  both  directions. 

Let  Te,  Tw,  Tw',  Te'  =  the  time  of  leaving  E  and  arriving  at 

W,  leaving  W  and  arriving  at  E, 
respectively; 

Ae,  Aw,  Awf,  Aef  =  the  corresponding  chronometer  cor- 
rections found  by  observation ; 
m  =  the  daily  travelling  rate. 
Then 

(Tw—  Te)-\-(Te'—  TJ)  =  time  during  which  the  chronometer 

was  in  transit ; 
(Af'—A^—(Aw'—  Aw]  =  the    corresponding    change    in    the 

chronometer  correction ; 

(Ae'-Ae}-(Aw'-Aw) 
-(Tw-Te)+(Te'-TwJ 

Previous  to  the  application  of  the  telegraph  to  the  deter- 
mination of  longitude,  the  construction  of  chronometers  had 
been  brought  to  such  a  degree  of  perfection  that  the  chro- 
nometric  method  was  the  most  accurate  one  available. 
Where  great  accuracy  was  required,  large  numbers  of  chro- 
nometers were  transported  many  times  in  both  directions. 
A  most  elaborate  expedition  of  this  kind  was  carried  out  in 
1843,  by  Struve,  for  determining  the  difference  of  longitude 
between  Pulkova  and  Altona.  Sixty-eight  chronometers 
were  carried  nine  times  from  Pulkova  to  Altona  and  eight 
times  from  Altona  to  Pulkova.  A  similar  expedition,*  or 

*See  Report  U.  S.  Coast  Survey,  1853,  p.  88;   1854.  p.  139;   1856,  p    182. 


382  PRACTICAL   ASTRONOMY.  §  214. 

series  of  expeditions,  was  conducted  by  the  U.  S.  Coast  Sur- 
vey during  the  years  1849,  '5°>  '5r>  an<^  '55>  m  which  fifty 
chronometers  were  transported  many  times  between  Boston 
and  Liverpool.  The  results  of  the  expeditions  in  the  years  '49, 
'50,  and '51  showed  the  necessity  of  introducing  a  correc- 
tion for  change  of  temperature.  The  expedition  of  '55  was 
therefore  planned  and  carried  out  under  the  direction  of  Mr. 
W.  C.  Bond,  with  special  reference  to  this  correction.  In 
this  year  fifty-two  chronometers  were  transported  three 
times  in  each  direction,  giving  as  the  difference  of  longitude 
between  the  Cambridge  observatory  and  the  observatory  at 
Liverpool — 

Voyages  from  Liverpool  to  Cambridge,  4h  32™  3i8-92  ; 
Voyages  from  Cambridge  to  Liverpool,  4  32    31  .75. 

Such  expeditions  are  enormously  expensive,  and  the  re- 
sults are  not  comparable  for  accuracy  with  those  obtained 
by  the  telegraph.  As  almost  every  point  of  much  importance 
on  the  habitable  part  of  the  earth  is  now  or  will  soon  be  sup- 
plied w.ith  telegraphic  facilities,  chronometric  expeditions  on 
the  scale  of  those  mentioned  may  be  reckoned  as  things 
of  the  past.  Nevertheless  the  chronometric  method  is  very 
useful  where  extreme  precision  is  not  required,  or  where  the 
telegraph  cannot  be  used,  as  at  sea. 

The  method  of  conducting  a  chronometric  expedition  is 
briefly  as  follows :  The  chronometers  at  the  first  station, 
which  we  may  suppose  to  be  E,  are  first  carefully  compared 
with  the  standard  clock ;  then  they  are  placed  in  the  vessel, 
near  the  middle  where  the  motion  will  be  the  least  possible, 
and  in  a  position  where  they  will  be  accessible  for  winding 
and  comparing  during  the  voyage.  They  should  be  com- 
pared daily  as  a  check  on  the  regularity  of  their  rates.  A 
record  of  the  temperature  must  be  kept. 


§215.  LONGITUDE  ^BY  CHRONOMETERS.  383 

On  arriving  at  W  the  chronometers  are  immediately  com- 
pared with  the  standard  clock  as  before  at  E. 

215.  The  errors  to  which  the  chronometers  are  liable  are 
of  two  kinds:  first,  accidental  irregularities  which  follow  no 
law  and  are  therefore  equally  liable  to  affect  the  result  with 
the  plus  or  minus  sign  —  the  larger  the  number  of  chronome- 
ters the  more  effectually  these  will  be  eliminated;  and  secondly, 
errors  resulting  from  acceleration  or  retardation  of  rate. 
When  the  chronometer  has  been  transported  a  number  of 
times  in  both  directions  the  effect  of  a  constant  acceleration 
or  retardation  may  be  eliminated  by  reckoning  the  longitude 
alternately  from  each  station  E  and  W. 

Experiments  show  the  acceleration  or  retardation  of  rate 
to  be  due  to  two  causes,  viz.,  Changes  of  temperature  and 
the  gradual  thickening  of  the  lubricating  oil.  This  latter 
diminishes  the  amplitude  of  the  vibration  and  therefore 
causes  an  acceleration  of  rate.  Its  effect  is  sensibly  propor- 
tional to  the  time. 

Although  great  care  is  given  by  the  makers  to  compensat- 
ing the  balance  for  temperature,  it  is  seldom  possible  to  ac- 
complish this  perfectly.  It  has  been  found  that  the  effect  of 
changes  of  temperature  may  be  represented  by  a  term  of  the 
form  k(%  —  3-0)2,  in  which  $9  is  the  temperature  of  rngst  per- 
fect compensation  and  3"  that  of  actual  exposure,  and  k  is  a 
constant  which  with  rare  exceptions  is  positive;  that  is,  ex- 
posure to  a  temperature  above  or  below  that  of  most  perfect 
compensation  causes  the  chronometer  to  run  slower. 

The  rate  of  any  chronometer  may  therefore  be  expressed 
by  the  formula 

$y-£'t;    ....     (380) 


kf  being  a  constant  depending  on  the  thickening  of  the  oil, 
or  any  other  causes  which  may  be  assumed  to  vary  directly 
with  the  time. 


384  PRA  CTICAL  A  STRONOM  Y.  §  2  1 6. 

The  constants  k,  k',  and  £0  peculiar  to  each  chronometer 
can  only  be  determined  experimentally. 

216.  The  term  depending  on  the  temperature,  k($0  —  5)2, 
having  always  the  same  sign,  will  never  vanish;  therefore  in 
order  to  find  the  total  effect  of  such  changes  during  any  in- 
terval a  strict  theory  requires  the  total  sum  of  all  these 
terms  for  all  changes  of  temperature. 

We  may  proceed  as  follows : 

Let  r  =  the  interval  during  which  the  effect  of  rate  is  re- 
quired ; 

Let  u0  of  formula  (380)  be  taken  at  the  middle  of  this  in- 
terval ; 

Let  r  be  supposed   divided  into  n  equal  parts,  so  small 

that   the   temperature   during  the   interval   —  may  be  con- 
sidered constant ; 

Let  3^,  £3,  . .  .  S*n  be  the  values  of  3"  for  each  interval  in 
succession. 

Then  the  accumulated  rate  for  each  interval  will  be  as 
follows  : 


-  (38i) 


§  2l6.  LONGITUDE  BY  CHRONOMETERS.  38$ 

The  sum  of  all  these  quantities  is  the  total  effect  of  rate ; 
and  as  the  sum  of  the  coefficients  of  kf  is  zero,  the  value  is 

u0r  +  tsfc  -  *.)'£  .     ....     (382) 

For  rigorous  accuracy  the  intervals  should  be  infinitesimal ; 
—  would  then  be  */r,  and  the  above  expression  would  be 


As,  however,  (3  —  30)  cannot  be  expressed  as  a  function  of  r, 
the  integration  is  not  possible. 

For  determining  ^"(3  —  30)2  we  write  the  mean  of  the  ob- 
served temperatures  (supposed  to  be  the  quantities  repre- 
sented above  by  3,,  32,  etc.)  equal  to  6. 

Then 


=  X;(^30)'+  %:  2(6  -  *.)($  -  6) 
Since  6  and  S0  are  both  constant,  we  have 

2lB  -  S.)'  =  n(6  -  ».)•;   ....     (383) 
2;  2(6  -  S.)  (S  -  6)  =  2(6  -  S0)^(5  -  6)  =  o,    (384) 


since  #  is  the  mean  of  the  individual  values  of  3. 
Therefore  (382)  becomes 

uer  +  k(9  -  S.)'r  +  ^0"(5  -  6>)'J     .     .     (385) 

^•n/  (X    _      /J\2 

The  value  of  the  quantity  -  -  is  computed  directly, 


since  S  is  any  observed  temperature,  and  8  the  mean  of  all 


336  PRACTICAL   ASTRONOMY.  §  2  1  8. 

the  values  observed.     This  will  approach  more  nearly  the 
theoretically  exact  value  the  more  frequently  the  tempera- 
tures are  observed. 
Writing 

*      -  ay 

=  *>   ......    (386) 


we  have  for  the  accumulated  rate  during  the  interval  r 

sjM-**F  .....   (387) 


The  quantity  in  the  brackets  is  the  mean  rate  during  the 
interval  r. 

217,  In   the  Coast   Survey  expedition  of    1855   tne  mean 
temperature   was  indicated  by  a  chronometer   constructed 
expressly  for  this  purpose.     It  was  in  all  respects  like  one  of 
the  ordinary  chronometers,  except  that  the  arms  and  rim  of 
the  balance  were  of  brass  and  uncompensated.     Its  indica- 
tions of  the  mean  temperature  of  exposure  were  found  to  be 
much  more  reliable  than  could  be  obtained  by  the  use  of 
ordinary  thermometers;    its  sensitiveness  was  such   that  a 
change  of  i°  in  the  temperature  produced  a  change  of  6S.5 
in  the  daily  rate.     Experiments  made  for  determining  the 
time  required  for  a  chronometer  to  adapt  itself  to  the  tem- 
perature of  the  surrounding  air  when  exposed  to  a  sudden 
change  showed  that  this  was  not  fully  accomplished  until 
five  or  six  hours  had  elapsed,  so  that  in  case  of  sudden 
changes  the  temperature  shown  by  the  thermometer  might 
differ  widely  from  the  actual  temperature  of  the  chronome- 
ter balance. 

218.  In  applying  (387)  to  any  subsequent  interval,  T',  u0 
.   must  be  replaced  by  u0  —  k't,  in  which  /  is  the  time  from  the 

middle  of  the  interval  r  to  the  middle  of  T'  '. 

Now  suppose  the  chronometer  used  for  determining  the 


§  2 1 8.  LONGITUDE  BY  CHRONOMETERS.  387 

difference  of  longitude  of  two  stations  E  and  W.  Suppose 
the  corrections  ^  and  ^2  determined  at  E  before  starting,  at 
the  times  7^  and  Tv  and  49  and  J4  after  reaching  W,  at 
times  7*3  and  T^  all  being  reckoned  from  the  same  meridian, 
suppose  E. 

Let  7;  -  T;  -  TI;      T;  -  7;  -  r,;      T;  -  T;  =  *,. 

rl  and  r3  are  shore  intervals,  and  r2  a  sea  interval. 

Let    UQ  =  the  rate  at  the  middle  of  the  sea  interval ; 
A  —  the  difference  of  longitude. 

Then  from  what  precedes  we  have 


-  (388) 


0j,  0a,  and  6Z  are  the  mean  temperatures  for  the  intervals, 
€lt  fa,  and  fs  having  the  values  given  by  (386).  Then  from 
the  three  equations  (388)  «0,  k ',  and  A  may  be  determined. 


Let  us  write     /   = 


-  (389) 


We  then  find,  from  the  first  and  third  of  (388), 

-^    (390) 


388  PRACTICAL   ASTRONOMY.  §  2I9- 

These  values  substituted  in  the  second  of  (388)  give  the 
value  of  the  longitude  A. 

The  chronometric  method  finds  its  most  important  appli- 
cation at  sea,  where  a  high  degree  of  precision  is  not  impor- 
tant. When  the  time  from  port  is  not  very  great,  this  will 
answer  all  practical  requirements.  When  the  voyage  is  very 
long,  the  result  may  be  rendered  much  more  accurate  by 
applying  the  corrections  for  acceleration  of  rate,  the  con- 
stants k,  k\  and  3-0  having  been  carefully  determined  pre- 
viously. 


Determination  of  Longitude  by  the  Electric  Telegraph. 

219.  The  local  time  at  one  meridian  may  be  compared  with 
that  at  another  most  conveniently  and  accurately  by  tele- 
graphic signals. 

The  most  simple  method  of  making  this  comparison  is  as 
follows :  The  operator  at  one  station  taps  the  signal  key  in 
coincidence  with  the  beat  of  the  chronometer;  the  instant 
when  the  signal  is  received  at  the  other  station  is  noted  by 
the  chronometer  at  that  place.  A  number  of  arbitrary  sig- 
nals are  sent  in  this  way,  when  the  process  is  reversed,  the 
operator  at  the  second  station  sending  the  signals  to  the 
first.  The  errors  of  the  chronometers  will  generally  be 
determined  by  observing  transits  both  before  and  after  ex- 
changing the  signals. 

Let  Te  and  ATe  =  the  chronometer  time  and  correction 
at  station  E  at  the  instant  of  sending 
a  signal ; 

Tw  and  ATW  =.  the  chronometer  time  and  correction 
at  station  W  at  the  instant  of  receiv- 
ing this  signal ; 


§219-      LONGITUDE  BY   THE  ELECTRIC    TELEGRAPH.  389 

TV  and  A  TV  =  the  chronometer  time  and  correction 
at  station  W  at  the  instant  of  sending 
a  return  signal ; 

TV  and  ATef  =  the  chronometer  time  and  correction 
at  station  E  at  the  instant  of  receiv- 
ing this  signal ; 

A  =  the  difference  of  longitude  ; 
p  —  the  transmission  time  of  the  electric 
effect,  or  the  small  interval  of  time 
which  elapses  between  the  instant  of 
pressing  the  key  at  one  station  and 
the  click  of  the  magnet  at  the  other. 

Then      A  -  p  =  (Te  +  ATe)  -  (Tw  +  ATW]  =  Ae; 

Therefore  A  =  -J(AW  -f  AY  \ 

u  =  ±(*       -AY (39I) 


Thus  by  eliminating  the  time  required  for  transmission  of 
signals  we  have  the  longitude,  or  by  eliminating  the  longi- 
tude we  have  the  transmission  time. 

For  many  purposes  the  above  process  will  give  a  sufficient 
degree  of  accuracy.  For  first-class  longitudes,  however, 
there  are  a  number  of  small  errors  involved  which  will  de- 
mand attention.  They  are  as  follows  : 

I.  The  relative  personal  equation  of  the  observers  in  deter- 
mining the  chronometer  corrections  at  the  two  stations. 
II.  The  personal  equations  involved  in  sending  and  receiv- 
ing the  signals. 

III.  The  time  required  at  the  sending  station  to  complete 

the  circuit  after  the  finger  touches  the  key. 

IV.  The  time  required  at  the  receiving  station  for  the  arma- 

ture to  move  through  the  space  in  which  it  plays  and 
give  the  click — called  the  armature  time. 


39O  PRACTICAL   ASTRONOMY.  §  222. 

If  the  two  latter  could  be  assumed  to  be  the  same  at  both 
stations,  the  above  errors  would  be  reduced  simply  to  per- 
sonal errors.  We  shall  describe  some  of  the  methods  of 
dealing  with  these  quantities  in  first-class  longitudes.  They 
may  be  modified  when  a  less  degree  of  accuracy  is  demanded. 

220.  I.  Personal  equation.     This  may  be  determined  by  any 
of  the  methods  given  in  Art.  188,  and  the  necessary  correction 
applied.     If  the  relative  personal  equation  is  used,  it  should 
be  determined  both  before  and  after  the  longitude  work  in 
order  to  guard  against  the  effect  of  its  gradual  change.     The 
plan  followed  by  the  Coast  Survey  is  to  exchange  signals  on 
five  nights,  then  let  the  observers  exchange  stations,  when 
signals  are  exchanged  on  five  more   nights.     The  personal 
equation  is  thus  eliminated,  provided  it  has  remained  constant 
during  the  time  emplo}7ed.     As  this  changes  with  the  physi- 
cal condition  of  the  observer,  its  variation  is  probably  the 
chief  cause  of  discrepancy  in  first-class  longitudes. 

221.  Errors  II  and  III  are  avoided  by  using  the  chrono- 
graph.     For    field-work     break-circuit    chronometers    will 
generally  be  used,  as  they  are  much  more  convenient  to 
carry  than  clocks.     Such  a  chronometer  being  placed  in  the 
circuit  may  be  made  to  record  its  beats  on  the  chronographs 
at   both  stations.     Each    chronograph    will   then    contain  a 
record  of  the  beats  of  both  chronometers,  the  mean  of  which 
will  be  free  from  the  transmission  time,  but  will  be  affected 
by  any  constant   difference   in  the   armature  time,  viz.,  IV 
above. 

222.  Another  method  of  sending  the  signals  is  the  follow- 
ing :  The  circuit  is  so  arranged  that  a  tap  made  on  the  signal 
key  at  either  station  is  recorded  on  the  chronographs  at  both 
stations.     The  observer  at  E  then  gives  a  number  of  taps  at 
intervals  of  two  or  three  seconds,  which  are  recorded  at  both 
places  in  connection  with  the  beats  of  the  respective  chro- 
nometers, when  the  operation  is  repeated  by  the  observer  at 


§  223.      LONGITUDE   BY   THE  ELECTRIC   TELEGRAPH.  39! 

W.  For  identifying1  the  hour  and  minute  of  difference  of 
longitude,  the  observer  at  each  station  informs  the  one  at  the 
other  by  a  telegraphic  message  what  was  the  hour,  minute, 
and  second  by  his  chronometer  when  the  first  signal  was 
sent.  The  hour  and  minute  of  one  signal  being  identified, 
only  the  seconds  and  fractional  parts  of  the  same  need  be 
read  for  the  remaining  signals. 

223.  IV.  The  armature  time  will  be  practically  the  same 
at  both  stations,  and  consequently  the  effect  will  be  elimi- 
nated if  the  resistance  of  the  line  is  kept  at  the  same  value  at 
both  points.  For  this  purpose  a  rheostat  and  galvanometer 
are  provided  at  both  stations,  by  means  of  which  the  resist- 
ance may  be  maintained  at  any  required  value. 

The  chronometer  is  placed  in  a  local  circuit  acting  on  a 
relay,  the  intensity  of  the  current  in  the  main  line  being  too 
great  for  the  delicate  mechanism  of  these  instruments. 

The  details  will  be  understood  by  reference  to  the  follow- 
ing diagrams,  taken  from  a  paper  by  Mr.  C.  A.  Schott.* 

I  shows  a  simple  circuit  for  observing  transits.     The  chro- 
nometer breaks  the  circuit  B,  causing  the  pen  on  the  arma- 
ture of  the  chronograph  magnet  to  record.     The  observer 
breaks  the  circuit  with  the  observing  key,  also  making  a 
record  on  the  chronograph. 

II  and  III  show  the  arrangement  of  the  circuit  for  chro- 
nometer signals:  II  being  at  the  sending  station,  III  at  the 
receiving  station.      When  the  chronometer  at  the  sending 
station  breaks  the  circuit  B,  the  armature  of  the  chronograph 
magnet  breaks  the  main  circuit  at  X  (II),  and  the  armature 
of  the  signal  relay  at  the  receiving  station  breaks  the  circuit 
B  (III)r  causing  a  record  to  be  made  on  the  chronograph. 

For  sending  arbitrary  signals  the  arrangement  is  the  same 
at  both  stations,  viz.,  that  shown  in  III.  At  the  sending 

*  Appendix  No.  14,  U.  S.  Coast  Survey  Report  1880. 


392 


PRACTICAL  ASTRONOMY. 


§223, 


Chronometer 


I. 


Chronograph  Magnet 
with  pen  on  armature 


Observing  Key  at  Transit 


II. 


Chronometer 


Talking  and 
Signal  Rela 

Main / 


—X 


Observing  Key  ht  Transit 


Talking  Key 


I    j 


__y  V,___^2l 


III. 


Chronometer 


Chronograph  Magnet 
with  pen  on  armature 


Observing  Key  at  Transit 


Line 


Talking  and  Signal  Key 


FIG. 


§224.      LONGITUDE  BY   THE  ELECTRIC   TELEGRAPH.          393 

station  the  main  circuit  is  broken  by  the  signal  key,  when  the 
armature  of  the  signal  relay  breaks  the  circuit  B  at  both 
stations,  causing  a  record  to  be  made  on  the  chronograph. 

In  these  cases  the  chronometer  is  placed  directly  in  the 
circuit  passing  to  the  chronograph,  and  no  provision  is  made 
for  equalizing  the  resistance  at  the  two  stations.'  A  smal/ 
difference  in  the  armature  time  is  therefore  likely  to  exist. 

Chronometer  IV. 

Battery  1C  ell 


Magnet 
pen  on  armature 


Observing  Key  at  Transit 

FIG.  44. 

224.  IV,  VII,  and  VIII  show  a  more  complete  arrange 
ment  of  circuits.  The  chronometer  is  placed  in  a  local  cir- 
cuit A  with  a  weak  battery,  in  order  to  avoid  the  injurious 
effect  of  a  stronger  current  on  the  mechanism.  When  ob- 
serving transits  the  arrangement  is  as  shown  in  IV.  The 

o  o 

chronometer    breaks  the  circuit  A,  the  chronometer  relay 
breaks  the  circuit  B,  making  a  record  on  the  chronograph. 

The  observer  breaks  circuit  B  with  the  observing  key, 
also  producing  record  on  chronograph. 

VII  shows  the  arrangement  for  exchanging  chronometer 
signals,   being  alike   at   both    stations.      The   chronometer 
breaks  circuit  A,  when  the  armature  of  the  chronometer  re- 
lay breaks  the  main  circuit,  the  armature  of  relay  D  break- 
ing circuit  B  at  both  stations. 

VIII  is  arranged  for  arbitrary  signals,  both  stations  being 
the  same.     The  chronometer  breaks  circuit  A,  the  armature 
of  chronometer  relay  breaks  circuit  B,  making  record  on  the 
chronograph.     At  the  sending   station  the  main  circuit  is 
broken  bv  the  signal  key,  when  relay  D  breaks  circuit  B  at 
both  stations. 


394 


PRACTICAL   ASTRONOMY. 


§224. 


Main  Circuit        

Local  Circuit 

.Arrangement  during  1881 
FIG.  45. 


§22$.      LONGITUDE  BY   THE  ELECTRIC   TELEGRAPH.  395 

By  means  of  the  rheostat  and  galvanometer  the  electric 
resistance  is  kept  practically  the  .same  at  both  stations,  and 
therefore  a  constant  difference  of  armature  time  avoided. 
In  order  to  eliminate  any  small  outstanding-  difference  in  the 
action  of  the  two  sets  of  electric  apparatus,  each  set  may  be 
used  at  both  stations  alternately,  the  instruments  being  ex- 
changed with  the  observers  at  the  middle  of  the  series. 

225.  Method  of  Star  Signals.  This  method  of  exchanging 
longitude  signals  was  formerly  employed  by  the  Coast  Sur- 
vey. A  very  full  description  of  the  method  is  given  by 
Chauvenet  (Spherical  and  Practical  Astronomy).  It  is  briefly 
as  follows : 

The  difference  of  longitude  between  two  points,  being 
simply  the  time  required  for  a  star  to  pass  from  the  meridian 
of  the  east  to  that  of  the  west  station,  may  be  measured  by  a 
single  clock  placed  in  the  electric  circuit  so  as  to  produce  a 
record  on  the  chronographs  at  both  points.  This  clock  may 
be  at  either  point,  or  in  fact  anywhere  in  the  circuit. 

When  a  star  enters  the  field  of  the  transit  instrument  at 
E,  the  observer  records  the  transit  by  tapping  his  signal  key 
in  the  usual  manner,  producing  a  record  on  both  chrono- 
graphs. When  this  star  reaches  the  meridian  of  W,  the  ob- 
server in  like  manner  taps  its  passage  over  the  threads  of 
his  transit  instrument,  also  producing  a  record  at  both 
points. 

This  method  is  theoretically  very  perfect;  but  as  it  requires 
a  monopoly  of  the  telegraph  lines  for  several  hours  every 
night  when  signals  are  exchanged,  it  has  proved  somewhat 
impracticable.  • 

Example. 

For  the  purpose  of  illustrating  this  subject  I  give  below 
the  record  of  a  series  of  longitude  signals  between  Washing- 
ton, D.  C.,  and  Wilkes  Barre,  Penn.,  1881,  October  6th. 


396  PRACTICAL   ASTRONOMY.  §  225. 

At  Washington  the  instruments  employed  were  the  tran- 
sit circle,  sidereal  clock,  and  chronograph  of  the  U.S.  Naval 
Observatory. 

At  Wilkes  Barre  the  instruments  were  a  portable  transit 
and  mean  time  chronometer. 

At  the  latter  place  the  following  programme  was  followed: 
Transits  of  16  stars  were  observed,  the  instrument  being 
twice  reversed ;  the  chronometer  was  then  taken  to  the 
telegraph  office,  200  feet  distant,  and  the  longitude  signals 
exchanged,  after  which  13  stars  were  observed  with  the  tran- 
sit instrument,  the  axis  being  reversed  once.  The  29  equa- 
tions furnished  by  the  observed  transits  gave  the  values  of 
the  chronometer  correction  and  rate,  also  the  azimuth  and 
collimation  constants  of  the  transit  instrument. 

The  following  is  the  method  adopted  in  exchanging  sig- 
nals : 

At  Washington  the  telegraph  ke,y  was  tapped  at  intervals 
of  about  15  seconds,  making  a  record  on  the  Washington 
chronograph,  and  through  the  telegraph  line  a  click  of  the 
sounder  at  Wilkes  Barre.  The  observer  at  the  latter  place, 
having  his  eye  on  the  chronometer,  noted  the  instant  of  this 
click  and  recorded  the  same.  After  10  or  15  such  signals  had 
been  sent  from  Washington  to  Wilkes  Barre,  a  similar  series 
was  sent  in  the  opposite  direction,  the  operator  at  Wilkes 
Barre  tapping  the  key,  producing  a  click  of  the  sounder  at 
that  place  and  a  record  on  the  Washington  chronograph. 

This  constitutes  a  complete  series.  Two  such  were  ex- 
changed each  night  when  observations  were  made. 

It  is  obvious  that  with* a  chronograph  at  Wilkes  Barre  noth- 
ing need  be  changed  in  the  above  programme.  The  record 
would  then  be  made  on  the  chronograph  instead  of  by  the 
observer,  and  if  thought  desirable  the  intervals  between  the 
signals  could  be  much  shortened. 

The  chronometer  at  Wilkes  Barre  being  regulated  to  mean 


§  225.      LONGITUDE  BY   THE  ELECTRIC   TELEGRAPH. 


397 


solar  time,  its  correction  and  rate  on  sidereal  time  are  some- 
what large.  The  values  obtained  from  the  observed  transits 
are  as  follo\*s : 

At  9h  39m  chronometer  time,    AT  —  +  !3h  9™  38S.9<D3  ±  .024 
Hourly  rate,  +     9 .952 

Rate  per  minute,  -\-        .1659 

Similarly  for  the  Washington  clock, 

At  22h  30™  sidereal  time,  AT  —  -  21  ".891  ±  .019 

Hourly  rate,  +       .0360 

The  record  of  the  signals  with  the  individual  values  of  the 
longitude  immediately  follows : 


Washington  to  Wilkes  Barre. 


No. 

Wilkes  B. 
chronome- 
ter. 

AT. 

Washington 
clock. 

AT. 

Wilkes  B. 
sidereal 
time. 

Washington 
sidereal 
time. 

Differ- 
ence of 
longitude. 

•v. 

i 

9h  39m  1  3"  -9 

I3h  9m  388.94 

22h44m  34B-44 

-  2i».88 

22h48m528.84 

22h44m  12s.  56 

4m  4o8.  28 

•03 

2 

39     28  .9 

38.98 

44    49  -So 

49      7  .88 

44    27  .62 

40  .26 

.01 

3 

39     43  -8 

39  -02 

45      4  -40 

40      22   .82 

44    42  .52 

4o  .30  - 

•05 

4 

39     58  -8 

39.06 

45    19  -38 

49    37-86 

44    57  -5° 

40  .36 

.16 

5 

40     13  .6 

39  -i° 

45    34  -30 

49    52  .70 

45     12  .42 

40   28 

•°3 

6 

40     28  .5 

39  -f4 

45    49  -28 

50      7  .64 

45    27  .40 

40  24 

.O[ 

7 

40    -n  -5 

39.18 

46      4  .32 

50    22  .68 

45    42  -44 

40   24 

.01 

8 

40    58  .8 

39  -23 

46    19  .56 

50    38  -°3 

45    57-6? 

40    35 

.10 

9 

41     13  .6 

39  -27 

46    34.66 

50    52  .87 

46      12  .78 

40   09 

.16 

10 

9    41     28  .6 

13    9     39  -3i 

22  46    49  .66 

-  21  .88 

22    5I         7  .91 

22    46      27  .78 

4     40    T3 

.12 

Mean  =  4    40  .253  =  Af 


Wilkes  Barre  to  Washingtc 


I 

9h  45m  iis.i 

13h  9m  39s.93 

22h50m  32s.  78 

—  2i8.88 

22h54m5is  03 

22h50m  10s.  90 

4m  4o8  13 

.09 

2 

45  26.1 

39-97 

50  47  .72 

55   6  .07 

50  25  .84 

40.23 

.01 

3 

45  3&  -° 

39  -99 

50  57  -74 

55  1.5  -99 

50  35  -86 

40  .33 

09 

4. 

45  5i  -i 

40  .04    51  12  .70 

55  3i  -!4 

50  50  .82 

40  .72 

.10 

5 

46   6  .4 

40  .08    51  28  .10 

55  46  .48 

51    6  .52 

40  a5 

.04 

6 

46  20  .7 

40  .12!   51  42  .52 

56   o  .82 

51   20  .64 

40  .18 

.04 

7 

46  35  -9 

40.16    51  57.62 

56  16  .06 

5i  35  -74 

40.32 

.10 

8 

46  50  .8 

4O  .20;     52   12  .66 

56  31  .00 

51  5«  -7& 

40  .22 

.00 

9 

47   6.1 

40  .25!   52  28  .  10 

56  46  -35 

52   6  .22 

40  .13 

.09 

10 

9  47  21  .1 

13  9  40  .29  22  52  43  .00 

-  21  .88 

22  57   i  .39 

22  52   21  .  12 

4  40  .27 

•05 

Alean  =  4    40  .219  =  Ae 


398  PRACTICAL  ASTRONOMY.  §  226. 

Then  referring  to  formulae  (391),  we  have 


A.  =  \(\w  +  Ae)  =  4™  40^.236  Wilkes  B.  east  of  Wn. 
V  =  ^w  -  **)=         0.017. 

In  the  above  the  reduction  of  each  signal  has  been  carried 
out  separately,  in  order  to  show  the  precision  of  the  individ- 
ual values.  Practically  the  labor  of  reduction  may  be  econ- 
omized by  reducing  the  means  of  the  recorded  times. 

Thus  from  the  above  we  have  — 


Wn.—  Wilkes  B. 

Wilkes  B.—  Wn. 

Wilkes 

Barre 

chronometer,  gh 

4Om  21s. 

20 

9h 

46- 

14s. 

53 

AT, 

13 

9 

39- 

13 

13 

9 

40. 

10 

Wilkes 

B.  sidereal 

time, 

22h 

50' 

11    o8. 

33 

22h 

55m 

54s. 

63 

Washington  clock, 

22 

45 

41  . 

95 

22 

51 

36. 

29 

4T, 

-  21  . 

88 

— 

21  . 

88 

Wn.  sidereal 

time, 

22h 

45' 

11  20s. 

07 

22h 

5im 

I4S. 

41 

Wn.—  Wilkes  B. 

Wilkes  B.—  Wn. 

Difference  of 

longitude  = 

4m  40s. 

26 

4m 

409.22 

A  =  4    40  .24  Wilkes  B.  east  of  Wn. 

JJl  =  .02 

This  value  of  A  is  affected  by  the  relative  personal  equa- 
tion of  the  observers  at  Washington  and  Wilkes  Barre,  by 
the  personal  equation  of  the  observer  at  Wilkes  Barre  in  re- 
cording the  signals,  and  by  the  difference  in  armature  time 
at  the  two  stations.  (See  Articles  220-223.) 

Longitude  Determined  by  the  Moon. 

226.  The  preceding  methods,  in  circumstances  where  they 
are  available,  leave  little  to  be  desired  in  facility  of  application 
or  in  accuracv  of  results.  Before  the  invention  of  the  electric 


§227-        LONGITUDE  DETERMINED  BY   THE  MOON.  399 

telegraph  the  most  valuable  methods  for  determining  longi- 
tude were  those  depending  on  the  moon's  motion,  chrono- 
metric  expeditions  being  generally  impracticable.  Though 
the  necessity  for  resorting  to  these  methods  is  constantly 
diminishing  as  the  telegraph  lines  become  more  widely 
extended,  it  will  probably  never  entirely  disappear. 

There  are  various  methods  of  utilizing  the  moon's  motion 
for  this  purpose,  the  most  important  of  which  are  the  follow- 
ing: 

By  eclipses  of  the  sun  and  occultations  of  stars. 

By  moon  culminations.  % 

By  lunar  distances. 

By  measurements  of  the  moon's  altitude  or  azimuth. 

Some  use  has  also  been  made  of  lunar  eclipses. 

All  of  these  methods  depend  upon  the  same  general  prin- 
ciple, viz. :  The  moon  has  a  comparatively  rapid  motion  of 
its  own,  in  consequence  of  which  it  makes  a  revolution 
about  the  earth  in  27^  days.  The  elements  of  its  orbit, 
together  with  the  effects  of  the  various  perturbing  forces, 
being  known,  it  is  possible  to  determine  the  position  of  the 
moon  at  any  given  instant  of  time ;  thus  in  the  American 
Ephemeris  and  Nautical  Almanac  will  be  found  the  right 
ascension  and  declination  of  the  moon  computed  several 
years  in  advance  for  every  hour  of  Greenwich  time.  Sup- 
pose now  at  a  point  whose  longitude  is  required  the  position 
of  the  moon  to  be  determined  in  any  convenient  manner  by 
observation ;  the  local  time  being  carefully  noted,  the  ephe- 
meris  above  mentioned  gives,  either  directly  or  through  the 
medium  of  a  more  or  less  extended  computation,  the  Green- 
wich time  corresponding  to  this  position.  A  .comparison  of 
this  Greenwich  time  with  the  observed  local  time  gives  the 
difference  of  longitude  required. 

227.  Some  of  the  applications  of  this  principle  are  capable 
of  giving  very  good  results  ;  but  there  is  one  difficulty  inhei- 


400  PRACTICAL   ASTRONOMY.  §  228. 

ent  in  the  principle  itself  which  precludes  the  attainment  of 
an  accuracy  commensurate  with  that  obtained  with  the  tele- 
graph.  The  angular  velocity  of  the  earth  on  its  axis,  which 
is  the  measure  of  time,  is  twenty-seven  times  greater  than  the 
angular  velocity  of  the  moon  in  its  orbit ;  it  follows,  there- 
fore, that  errors  of  observation  in  determining  the  moon's 
position,  or  of  the  ephemeris,  will  produce  errors  in  the 
resulting  longitude  twenty-seven  times  as  great.  So  if  the 
errors  to  be  anticipated  in  determining  the  place  of  the 
moon  are  of  the  same  order  as  those  of  determining  and 
comparing  the  errors  of , the  clocks  by  the  electric  telegraph, 
we  might  expect  to  attain  to  an  ultimate  degree  of  precision 
bv  the  latter  method  twenty-seven  times  greater  than  by  the 
former. 

Longitude  by  Lunar  Distances. 

228.  This  method  is  chiefly  useful  on  long  sea-voyages, 
where,  in  consequence  of  accumulating  errors,  the  indications 
of  the  chronometers  become  unreliable. 

The  observation  consists  in  measuring  with  a  sextant,  or 
other  suitable  instrument,  the  distance  of  the  moon's  limb 
from  that  of  the  sun,  or  from  a  neighboring  star,  the  time 
being  noted  by  the  chronometer.  After  this  measured  dis- 
tance has  received  the  necessary  corrections  (to  be  :onsid- 
ered  hereafter),  the  Greenwich  time  corresponding  is  taken 
from  the  tables  of  lunar  distances  of  the  ephemeris  by  the 
methods  of  Art.  55.  The  difference  between  this  time  and  the 
recorded  chronometer  time  is  the  error  of  the  chronometer 
on  Greenwich  time.  An  altitude  of  the  sun  or  a  star  gives 
the  error  on  local  time ;  the  difference  between  the  two 
errors  is  the  difference  of  longitude. 

The  ephemeris  gives  the  distance,  as  seen  from  the  centre 
of  the  earth,  of  the  moon's  centre  from  the  centre  of  the  sun, 


§  22Q.  LONGITUDE  BY  LUNAR  DISTANCES.  40! 

from  the  four  larger  planets,  and  from  certain  fixed  stars 
situated  approximately  in  the  path  of  the  moon.  They  are 
given  at  intervals  of  three  hours  Greenwich  mean  time. 

By  a  series  of  carefully  observed  lunar  distances  on  both 
sides  of  the  moon  the  chronometer  error  may  generally  be 
ascertained  within  twenty  or  thirty  seconds.  A  longitude 
determined  in  this  way  should  be  considered  as  liable  to  an 
error  of  five  miles,  a  degree  of  accuracy  which  answers  the 
requirements  of  navigation. 

229.  We  shall  consider  first  the  distance  of  the  sun  and 
moon. 

This  distance  having  been  measured  and  corrected  for  in- 
strumental  errors,  such  as  index  error  and  eccentricity,  the 
result  is  the  apparent  distance  between  the  limbs  of  the  sun 
and  moon  as  seen  from  the  point  of  observation.  In  order 
to  have  this  comparable  with  the  distances  of  the  ephemeris 
it  must  be  corrected  for  the  semidiameters,  parallaxes,  and 
refraction  of  the  two  bodies. 

In  order  to  apply  the  necessary  corrections  a  knowledge 
of  the  altitudes  at  the  time  of  observation  is  necessary. 
When  there  are  instruments  and  observers  enough,  which 
will  frequently  be  the  case  at  sea,  all  of  the  quantities  may 
be  observed  simultaneously  :  the  altitude  of  the  sun  so  ob- 
served, if  that  body  is  sufficiently  far  from  the  meridian,  may 
be  further  utilized  for  determining  the  local  time. 

When  it  is  not  expedient  to  make  all  these  measurements 
at  once  the  observer  may  measure  the  altitudes  of  the  sun 
and  moon  immediately  after  measuring  the  distance  between 
these  bodies,  the  altitudes  at  the  time  of  that  observation 
being  computed  by  assuming  the  change  in  altitude  to  be 
proportional  to  the  change  of  time,  an  assumption  which  will 
not  be  much  in  error  if  the  time  is  short. 

Finally,  the  altitudes  may  be  computed  by  formulae  (II), 
Art.  65,  the  right  ascensions  and  declinations  being  taken 


402  PRACTICAL  ASTRONOMY.  §  230. 

from  the  Nautical  Almanac.  The  apparent  altitudes  will  be 
derived  from  these  computed  values  by  applying  the  correc- 
tion for  refraction,  table  II,  and  parallax  formulas  (VI)  and 
(VI),,  Art.  8 1.  This  supposes  the  longitude  to  be  approxi- 
mately known ;  otherwise  we  lack  the  means  of  determining 
the  hour-angle  /,  required  in  formulas  (II):  but  we  shall 
always  be  in  possession  of  a  value  sufficiently  accurate  for  this 
purpose.  If  in  an  extreme  case  this  be  not  true,  we  may 
repeat  the  computation,  using  the  value  of  the  longitude  ob- 
tained from  the  first  computation  as  the  assumed  approximate 
value. 

The  corrections  necessary  to  apply  to  the  measured  dis- 
tance may  be  computed  as  follows. 


Correction  for  Semidiameter  of  Sun  and  Moon. 
230.  The  following  quantities  are  taken  from  the  epheme- 


ns: 


/=  the  geocentric  semidiameter  of  the  moon ; 
5  =  the  geocentric  semidiameter  of  the  sun ; 
n  =  the  equatorial  horizontal  parallax  of  the  moon ; 
Tl—  the  equatorial  horizontal  parallax  of  the  sun. 

The  moon  being  comparatively  near  the  earth,  the  semi- 
diameter  will  vary  appreciably  with  the  altitude ;  there  will 
be  no  appreciable  variation  in  the  case  of  the  sun.  The 
moon's  semidiameter  varies  inversely  as  the  distance. 

In  Fig.  46,  MOB  =  s. 

Call    MA  C  —  sf  —  apparent  semidiameter. 

s'  _  A     _  sin  MAZ _  sin  (Z  -\- 
Then  s  ==  jr=s\nMOZ-     ~sIn~Z 


LONGITUDE  BY  LUNAR  DISTANCES. 


403 


Z  being  the  geocentric  zenith  distance  of  the  moon,  and  /  the 
parallax  in  zenith  distance. 

sin  (Z+/)=sinZcos/+cosZsin/=sinZ-]-sin/cosZ,  nearly ; 
from  (128),          sin/  =  sin  n  sin  Z,  approximately. 
Therefore  s'  —  s(i  +  sin  n  cos  Z) (392) 


The  eccentricity  of  the  meridian  has  been  neglected,  but 
the  error  is  inappreciable  for  this  purpose. 

The  correction  for  semidiameter  will  be  still  further  modi- 
fied by  refraction.  Owing  to  this  cause  the  apparent  disks 
of  the  sun  and  moon  are  approximately  ellipses,  the  refrac- 
tion being  less  for  the  upper  limb  than  for  the  centre,  which 
in  turn  is  less  than  for  the  lower  limb.  We  therefore  require 
the  radius  of  the  ellipse  drawn  to  the  point  where  the  curve 
is  intersected  by  the  great  circle  joining  the  centres  of  the 
sun  and  moon. 


404 


PR  A  C  TIC  A  L   AS  TRONOM  Y. 


230. 


Regarding  the  figure  of  the  disk  as  an  ellipse,  the  conju- 
gate  axis    will   coincide    with   the   vertical    circle    passing 
through  the  centre,  the  semi-transverse 
axis  will  be  equal  to  s'  in  case  of  the 
moon ;    b,    the  semi-conjugate   axis,   is 
found  directly  from  the  refraction  table 
by  taking   out  the   refraction   for   the 
x  altitude  of  the  upper  and  lower  limbs 
respectively  and  subtracting  one  half 
the   difference  from  s '.      The  angle  q 
formed  by  the  radius  sq  with  the  con- 
jugate axis  is  the  angle  formed  with  the  vertical  circle   by 
the  great  circle  joining  the  centres  of  the  sun  and  moon ;  sq 
being  the  required  semidiameter. 
To  find  the  angle  q. 
In  the  triangle,  Fig.  48,  Z  is  the 

zenith;  J/and  S,  the  moon  and  sun.  /  VO-H 

Then 


FIG. 


cos  q  — 


sin  H  —  sin  //  cos  D 


cos  h  sin  D 

j 

—  

FIG.  48. 

/sin  \(D  -f-  h 

-  H)  cos  #i 

D  +  h- 

•f  H) 

sin  ±(D  —  h  +  H)  cos 


h  -  H) 


-     -     (693) 


For  computing  the  angle  at  the  sun,  h  and  H  will  be  inter- 
changed. 

Then  jn  the  ellipse  (Fig.  47)  we  have 


x  —  sq'  sin  q  ; 
y  =  sq'  cos  q  ; 


/y 


§231.  LONGITUDE  BY  L UNAR  DISTANCES.  405 

Therefore  s'  =  -—  (394) 

1/Y*  cosV  +  P  sin2? 

231.  The  values  of  sq'  computed  by  (394)  for  both  sun 
And  moon  are  then  to  be  applied  to  the  measured  distance 
of  the  limbs  of  those  bodies.  We  thus  have  the  measured 
distance  of  the  centres  as  seen  from  the  place  o4"  observation. 
To  obtain  the  required  geocentric  distance  this  must  now  be 
corrected  for  refraction  and  parallax. 


Let    D ',  H',  and  h'  —  the  apparent  distance  and  altitudes 

of  the  sun  and  moon  ; 

D,  H,  and  h  =  the  true  geocentric  distance  and  alti- 
tudes. 

H  and  h  are  obtained  by  applying  to  H'  and  h'  the  correc- 
tions for  refraction,  table  II  or  III,  and  for  parallax  formulae 
(VI)  and  (VI),,  Art.  81. 

Referring  to  Fig.  48, 

cosD'=s'\nff 'sink' -\-cosfi' cosh' cosE=cos(ff'— h')— cos/f'cosA^sin9^;  )  /      , 
cosD  =  sin  H  sink  -l~cos^cos^  cos£'=cos(^'  —  k)—coslf  cosh  2sin'*$E.  ) 


Multiplying  the  first  of  the  preceding  equations  by  cos  H 
cos  h,  and  the  second  by  cos  H'  cos  h',  then  subtracting  to 
eliminate  sin2  \E,  we  find 


/-*/-\c     - 

cos  D=cos  (H-h)  +  _r-  [cos  ZT-cos  (H'-k')}.  (396) 


D  is  therefore  expressed  in  terms  of  known  quantities.     The 
equation  is  not,  however,  in  convenient  form  for  numerical 


406  PRACTICAL  ASTRONOMY.  §  232. 

computation ;  therefore  we  make  the  following  transforma- 
tion: 


cos  H  cos  h    i     cos  D 
Let   -  TJ-,  ---  77  =  7=.  ;    —  -^  —  =  cos  D  ; 
cos  H'  cos  h'   C       C 


H  —h  =<*; 


*  (397) 


It  may  readily  be  shown  that  C  will  never  be  so  small  as 
to  give  impossible  values  to  D"  and  d"  . 
(396)  then  reduces  to 

cos  D  —  cos  D"  =  cos  d  —  cos  d"  ; 
from  which 

sin  \(D  -  D")  =  45M±         sin  «rf  -  O  ;  .     (398) 


and  with  accuracy  sufficient  for  practical  purposes, 


As  the  unknown  quantity  Z?  is  involved  in  the  second 
member,  this  equation  must  be  solved  by  approximation. 
Writing  in  the  denominator  D'  +  D"  for  D  +  D",  we  obtain 
a  value  of  D  which  will  generally  be  sufficiently  near  the 
true  one.  In  case  the  value  found  in  this  way  differs  very 
widely  from  D',  the  computation  may  be  repeated,  using  this 
value  just  found  in  the  denominator  of  (399). 

232.  In  the  above  we  have  assumed  the  angle  E  (the  dif- 
ference between  the  azimuth  of  the  sun  and  moon)  to  bt  the 


§232.  LONGITUDE  BY  LUNAR  DISTANCES.  407 

• 

same  for  the  point  of  observation  as  for  the  centre  of  the 
earth.  We  have  seen,  however,  that  the  moon  has  an  ap- 
preciable parallax  in  azimuth  the  value  of  which  is  given  by 
formulae  (VI),  Art.  81,  or  (VII),  Art.  82. 

In  order  to  determine'  the  correction  to  D  due  to  this 
quantity,  we  differentiate  the  second  of  (395)  with  respect  to 
D  and  E,  viz., 

cos  H  cos  h  sin  E  , 

dD  =  -       —-r      -  da>  •    '    •    '     (400) 


remembering  that  dE  =  da. 

da  is  the  parallax  in  azimuth  computed  by  the  formulae 
above  referred  to. 

Formulae  (392),  (393),  (394),  (397)»  (399);  (400)  now  give  the 
true  geocentric  distance  D,  corresponding  to  the  measured 
distance  D'.  Then  by  the  method  explained  in  Art,  55  we 
take  from  the  ephemeris  the  Greenwich  time  corresponding 
to  this  distance  ;  the  difference  between  this  time  and  the 
observed  time  will  then  be  the  chronometer  correction  on 
Greenwich  time. 

If  a  planet  has  been  used  instead  of  the  sun,  the  same 
formulae  will  be  used  ;  but  if,  as  is  generally  the  case,  the 
disk  of  the  planet  is  bisected  by  the  lirnb  of  the  moon  in 
making  the  observation,  there  will  be  no  correction  for  semi- 
diameter  of  planet.  The  effect  of  parallax  in  case  of  the 
outer  planets  will  be  very  small. 

If  the  distance  of  the  moon  from  a  star  is  measured,  there 
will  be  no  correction  for  semidiameter  or  parallax  of  the  star. 


408 


PRACTICAL   ASTRONOMY. 


§233- 


233.  Formula  for  Reducing  an  Observed  Lunar  Distance  to 
the  Geocentric  Distance. 


sr  =  s(i  -j-  sin  7t  cos  2)  ; 

rt 

3. 

1 

o> 
,  Art. 

(397) 
(399) 

c  ion  for 
allax    in 
nuth. 

fo«  1/7-      /sm  -k^D+h—H  )  cos  ^(D-\-k-\-H)^ 

tVa  cos*  ^  -|-  £2  sin*  ^' 

For  parallax  of  moon,  (VI),  Art.  8  1  ,  or  (VII) 
82. 
For  parallax  of  sun,  (VIII),,  Art.  82. 

cos  H  cos  h         i         cos  D'               n»r 

TTf                     11     ~S~*1                             S^               COS    J~J        . 

cos  H  cos  h        C            C 
H'-h>  =  d>-       cos  d' 

H   -  k  =d-,            C 

SlK±(d+d") 

~  sin  i(Z?  +  ZT)  (' 

COS  ^T  COS  h  Sin  £                     Corre 

^-ty  —                 .      ,-.              da.           Par 

SHI  D                                           azu 

(XXII) 


These  formulae  have  been  written  down  rigorously,  but  in 
practice  many  abridgments  may  generally  be  made  in  the 
application. 

Example.  1856,  March  gth,  5h  I4m  6s  local  mean  time,  the  following  distance 
of  the  nearest  limbs  and  altitudes  of  the  lower  limbs  of  the  sun  and  moon  were 
measured: 

D'  =  44°  36'  58".6;  H'  —  8°  56'  23'';  h'  =  52°  34  o". 

These  values  are  corrected  for  instrumental  errors. 

Barometer  29.5  inches;          Attached  thermometer  60°;          Detached  ther.  58°; 
Latitude  cp  —  35°;     Assumed  longitude  L  =  150°  =  ioh  west  of  Greenwich. 


233 


LONGITUDE  BY  L UNAR  DISTA NCES. 


409 


From  the  Nautical  Almanac  we  take  the  following  quantities: 

Sun.  Moon. 

Right  ascension,          a  =      23h  22m  27"  '  2h  nm  47* 

Declination,                   d  =  —  4°    3'     6"  14°  18'  41" 

Semidiameter,               S=             16      8  .o  s  =         16    23.1 

Horizontal  parallax,  II  =                      8  .6  n  •=.         60      I  .9 
Sidereal  time,  mean  noon,       23'*  nm    5s 

From  the  refraction  table  we  find,  for  the  altitudes  above  given, 


Refraction,  lower  limb, 
Approx.  altitude  of  centre, 


5  42  -9 

6'  48" 


43 

52°  49'  40 


We  now  compute  the  apparent  or  augmented  semidiameter  of  the  moon  by 
the  first  of  (XXII).  and  then  the  oblique  semidiameter  of  both  sun  and  moon 
by  the  second  and  third  of  these  formulae. 


z  =  37 
n  =    i 


10 
o' 


i".9 


s  =  983.1 
j'  =  996.8 

Measured  D'  =  44°  36'  58".  6 
16  36  .8 


S  = 

16     8  .0 

Approximate  D'  =  45 

9  43  -4 

Sun. 

O-  45°  10' 

&=  52    51 

h-     9    12 

$(D+h-H}=     o    45  . 

5     sin  =  8.i2i7 

HD+k+A 

0=  53    36 

cos=:9.7734 

\(D—h-\-h 

0=  44    25 

cosec=  .1550 

\(D—h  —  L 

0=-8    26 

sec=        47 

Sum  =  8.o548 

i 

\9=     6°    5' 

tan  ^—9.0274 

g=  12    10 

cos  z  —  9.9014 
sin  it  =  8.2419 
Sum  =  8.1433 

log  (i  -f-  sin  it  cos  z)  =  .0060 
log  =  2.9926 
log  =  2.9986 


Then  for  computing  q\ 

Moon. 

=  45°  10' 

12 


/y=     9 

h=  52 
O=  44 
53 
$(D- &-{-&)=     o 


=  79 


25 
36 
45  • 
26 

56' 

52 


cos  =9.  7734 


sec 
Sum 


47 
1.5014 

.7507 


Then  from  the  refraction  table  we  find — 

Refraction — upper  limb   = 

centre  —    5   33 

Therefore  b  —  15'  59  .2 


lower  limb  =        43".  i 

centre  =        42  .7 

b  —  16'  36". 4 


4io 


PRACTICAL   ASTRONOMY. 


§233 


log  b  =  2.9819      log£2  =  5.9638 

sin2  q  =  8.6476 

4.6114 

A*  =  1.3407 

log  S  —  2.9859     log  S1  =  5.9718 
cos*  <?  =  9.9803 

5.9521 

B*  =  1.3601  £*  —    .9267 

5.97I5  5-9971 

ac  log  den.  =7.0143      logd.  =  2.9857    ac  log  den.  =  7.0014     logd.  =  2.9986 

log  Sq  =  2.9821  log  Sq    =   2.9984 

Sq  =  15'  59".6  sq'  =  16'  36". 4 


log  b  —  2.9984       log£2  =  5.9968 

sin2  q  =  9.0736 

5.0704 

A*  -    .8720 
log  s'  =  2.9986     log/2  =  5.9972 

COS2  q  =  9.9452 


Obs.  Zy=44°36'  58. "6 
True  D'  =  45     9  34.  6 

An  approximate  value  of  the  azimuth  of  the  moon  is  required  for  computing 
the  parallax;  also  of  the  sun  for  computing  the  small  correction  dD  given  by 
the  last  of  (XXII).  The  formulae  for  this  computation  are  f 


tan  M  — 


tan  d 


tan  a  = 


cos  M 


tan 


cos  /  '  sin  (<p  —  M) 

Converting  the  mean  time  of  observation  into  sidereal  time  (Art.  94),  we  find 


0  =    4h  26m    38 
Sun  a  =  23    22    27 

/  =  (9  -  or)  =    5      3    36 

t  =  75°  54' 


Moon  a  =  2h  nm  47" 
/  =  2    14    16 

t  =  33°  34' 


005=9.3867 


Sun. 

*=-  4°    3'         . 
t=     75    54 
M=  — 16   12 

<7>r"T          *35          O 

<p—M=     51    12  cosec(«p— M)=  .1083 

cos  ^7=9.9824 

tan  /=  .6000 

a=     78°  29'  tana=  .6907 


Moon. 


=  14    19 
'=33    34 


tan =9.4067 


<P=3S      o 

>—  M=i"7°  59'  cosec((?>-^/)=  .5104 
cos  M=q.<)8o6 


^=64°    3' 


tan  a=   .3129 


*  Addition  logarithms. 


t  (II),  Art.  65. 


§233- 


LONGITUDE  B  Y  LUNAR  DISTANCES. 


411 


For  parallax,         (VIII)i,  Art.  82,  (VII),  Art.  82, 

e'  —  z  —  II  sin  z'\  y  =  (<p  —  cp")  cos  a. 

p  sin  TT  cos  (q>  —  <p")  sin  (z1  —  y) 
-  ~ 


sin  (z  —  z)  = 
sin  (a'  —  a)  = 


p  sin  it  sin  (<p  — 


sn 


/  =  80°  53'  n".4 
log  it  —  0.9345 
sin  0'  =  9.9945 
log  (z   —  z)  =  0.9290 

s'  —  z  —  8". 5 

Therefore  // =    9°  6' 57".! 


log  p  =  9-99952 

sin  7t  =  8.24208 

cos  (q>  —  (p'}  —  o 

sin  (z    —  y)  =  9.78036 

sec  y  =  o 

sin  (z'  —  z)  —  8.02196 

z   —  z  =         36'    9". 6 
h  =  53°  26'    3".3 


log  (q>  —  cp')  =  2.81158 
cos  a  =  9.64106 
log  Y  =  2.45264 
Y  —  4'  44" 

z'  =  37°  10'    6". 3 
z'  -  r  =  37      5   22 

log  p  =  9-99952 

sin  it  =  8.24208 

sin  (q>  —  <p)  =  7.49715 

sin  a    —  9.95384 

cosec  z  =    .22494 

sin  («'  —  a)  =  5.91753 

a!  —  a  =  17".! 


We  now  compute  (397)  ana  v399): 


H 

= 

9° 

6' 

57' 

'.i 

COS 

=  9-9944799 

h 

= 

53 

26 

3 

•  3 

COS 

=  9-7750603 

H' 

= 

9 

12 

22 

.2 

sec 

=  .0056304 

h' 

= 

52 

50 

36 

•  4 

sec 

=  .2189667 

d 

= 

—  44 
-  43 

!9 

38 

6 
14 

.2 
.2 

log  £. 

=  9-994I373 

cos  D'   —  9.8482718 
cos  D"  =  9.8424091 

cos  d'   =  9.8595724 
eos</"  =  9-8537097 


D'   =       45°  9'34"-6 

D"  -       45  55     7  -o 

d  =  —  44  19     6.2 

d"  =  -  44  26  13  .7 

£(</+</")  =  -  44  22  40  .0 

+  />")  =       45  32  21  .8 

d  —  d"  —  427.5 


412  PRACTICAL   ASTRONOMY.  §  233- 

First  Approximation.  Second  Approximation. 

sin  \(d  +  d  ")  =  9. 84472n  sin  |(</  +  </  ")  =  9. 84472^ 

log  (d  —  d")  =  2.63094  log  (d  —  </")=  2.63094 

cosec  \(D'  -f-  /?")  =    .14647  cosec  %(D  -f-  D")  =    .14409 

log  (D  -  D")  =  2.622i3n  log  (D  -  D"}  =  2.61975,, 

D  -  D"  =  -  418.9  D  -  D"  =       -  6'  56". 6 

D  =  45°  48'    8''.  D  —  45°  48'  io".4 

i(Z? +  £>")  =  45    5137.5  dD=                 3.5 

D  =  45    48   13  .9 
Correction  for  parallax  in  azimuth: 

E  =  A'  —  a  =  14°  26' 

cos  H  =  9.9945 

cos  h  —  9.7751 

sin  ,£"  =  9.3966 

cosec  D  =     .1445 

log  (a   —  a]  —  1.2330 

log</£>  =  0.5437 

dD=      3"- 5 

We  have  now  to  take  from  the  Nautical  Almanac  the  Greenwich  time  corre- 
sponding to  this  distance  by  the  method  explained  in  Art.  55.  For  1856,  March 
gth,  we  find  the  following  distances  of  the  sun  and  moon : 

I2h     D  =  43°  59'  3i"  PL  =  .2493  r 

15  45    40  54  .2510 

18  47    21   53  .2527  I7 

We  have  therefore  to  interpolate  between  ish  and  i8h. 
Referring  to  formula  (106),  we  have 

J'  =         7'  19". 9  log  =  2.6433 

PLA  =    .2510 

t  _          i3m    4s         log  /  =  2.8943 

Therefore  T=  15"  13™    4s 

*  Correction  for  2d  difference  —  i 

Resulting  Greenwich  time       15    13       3 

Local  time  of  observation         5    14      6 

Resulting  longitude         9    58     57 

The  above  solution  of  this  problem  is  only  one  among  many,  as  it  has  re- 
ceived much  attention  from  mathematicians  on  account  of  its  importance  to 

*  Taken  from  table  I  at  the  end  of  the  Nautical  Almanac. 


§235-  LONGITUDE  BY  MOON  CULMINATIONS.  413 

navigators.  The  majority  of  the  solutions  are  only  approximate,  the  design 
being  to  reduce  the  numerical  work  to  a  minimum  without  at  the  same  time 
sacrificing  too  much  in  the  way  of  accuracy.  Such  methods  may  be  found  in 
any  work  on  navigation,  and  will  be  preferred  where  only  an  approximate  so- 
lution is  required. 

As  may  be  seen,  the  solution  which  we  have  given  may  be  considerably 
abridged  without  a  great  sacrifice  of  accuracy.  The  differences  between  the 
oblique  and  vertical  semidiameters  of  the  sun  and  moon  are  very  small,  and  the 
correction  for  parallax  in  azimuth  is  not  large.  When  we  remember  that  the 
least  reading  of  the  sextant  is  10",  and  that  measurements  of  this  kind  are  quite 
difficult,  it  will  be  seen  that  often  little  will  be  lost  by  neglecting  this  part  of 
the  computation. 


Longitude  by  Moon  Culminations. 

234.  The  right  ascension  of  the  moon  may  be  determined 
by  means  of  a  transit  instrument,  mounted  at  the  place  whose 
longitude  is  required,  and  the  local  time  of  observation  com- 
pared with  the  Greenwich  time  corresponding  to  this  right 
ascension,  either  by  taking  this  time  from  the  ephcmeris  of 
the  moon,  or  by  means  of  similar  observations  made  at 
Greenwich,  or  some  place  whose  longitude  from  Greenwich 
is  known. 


Comparison  by  means  of  the  Ephemeris. 

235.  The  transit  instrument  having  been  adjusted  as  ac- 
curatelv  as  may  be,  the  transit  of  the  moon's  bright  limb  is 
observed,  together  with  a  number  of  stars  suitable  for  de- 
termining the  errors  of  the  instrument  and  the  clock  cor- 

o 

rection.  The  corrections  necessary  to  give  the  moon's  right 
ascension,  from  the  observed  time  of  transit  of  the  limb,  are 
then  applied  according  to  formulae  (XIX),  Art.  195.  The  last 
term  of  the  formula  may  be  taken  from  the  table  of  moon 
culminations  where  it  is  given  under  the  heading  "  Sidereal 
time  of  semidiameter  passing  meridian." 


4^4  PRACTICAL  ASTRONOMY  §  237. 

236.  To  insure  greater  accuracy,  the  moon's  right  ascen- 
sion may  be  derived  by  comparing  the  observed  time  of 
transit  with  that  of  about  four  stars  differing  but  little  from 
the  moon  in  declination,  two  culminating  before  the  moon 
and  two  after.  A  list  of  stars  suitable  for  this  purpose  was 
formerly  given  in  the  ephemeris,  under  the  heading  "  Moon 
culminating  stars,"  but  it  has  been  discontinued  since  1882.' 
It  is  an  easy  matter  for  the  observer  to  select  suitable  stars 
from  the  general  list  of  the  ephemeris. 

Let  A0  =  the  right  ascension  of  the  moon's  bright  limb  at 

the  instant  of  culmination  ; 
A  =  the  right  ascension  of  the  moon's  centre  ; 
0  —  clock  time  of  observed  transit  of  limb,  corrected 
for  all  known  instrumental  errors  and  for  rate; 
a  .  6  =  right  ascension  and  time  of  transit  respectively 
of   a   star,  the  time  being  corrected    for   in- 
strumental errors  and  rate  of  clock  ; 
Sj  =  sidereal  time  of  semidiameter  passing  the  meri- 
dian, taken  from  ephemeris. 

Then 

(401) 


This  quantity  A  is  then  the  local  sidereal  time  of  transit  of 
the  moon's  centre. 

237.  We  have  now  to  take  from  the  ephemeris  of  the  moon 
the  Greenwich  mean  time  T  corresponding  to  this  value  A 
of  the  moon's  right  ascension;  the  mean  time  T  must  then 
be  converted  into  the  corresponding  Greenwich  sidereal  time 
O8.  Then  A  being  the  difference  of  longitude,  we  have 

*  =  ©o  -  A  .......     (402) 


§237-  LONGITUDE  BY  MOON   CULMINATIONS.  415 

The  time  T  may  be  interpolated  to  second  differences  from 
the  ephemeris,  as  follows  : 

Let        Al  =  the  ephemeris  value  nearest  to  A\ 
Tt  =  the  corresponding  time. 

Then  Tl  -f-  t  —  the  required  time  corresponding  to  A. 


-    A  4.-'4.*i 

-  A  -*       ~f  2' 


Let  A  A  =  the  difference  of  right  ascension  for  I  minute, 

taken  from  the  ephemeris  ; 
dA  =  difference  between  two  consecutive  values  of 

A  A. 

dA  then  equals  the  change  in  A  A  in  one  hour.  Then  if  /  is 
supposed  expressed  in  seconds,  we  shall  have  to  second  dif- 
ferences inclusive 

<M    _  AA     ,    ^A.         8A_ 
dT  ~:  6o";        dT*  ~=  (6of 

~ 


A  =  A,+        AA  +        ..  , 

r6oL  2     3600  J 


_  6o[A  - 

From  which  t  —  — 


A  A  +  dA  —  . 
7200 

and  with  sufficient  accuracy, 

dA 

* 


60  [A  -  A,-] 
Writing      *--         L~^, 

then  (403)  becomes  /  =  x  -f-  x"  .......     (404) 


41 6  PRACTICAL   ASTRONOMY.  §  238. 

Example.  Among  the  observations  of  the  moon  made  at 
Washington  I  find  the  following: 

1877,  May  23d.     Observed  right  ascension 
of  the  moon's  centre,  A  =  i3h  28m  5S.O2 

From  ephemeris  of  the  moon,  7i  =  I4h,     Al  =  13   27     3  .91 
AA  =      2.0996  A  —  A1  =  i     i  .11 

dA  =.+  .0029        6o(A  —  A^  =  3666.6      log  =  3.56426 

log  A A  —    .32213 

x  =  29m  63.4  log  x  =  3.24213 

;r"  =          -  .6  log  JIT'  =  6.48426 

/  =  29    5  .8       log  (-  dA)  =  7-46240, 
ac  log  A A  —  9.67787 
aclog  7200  =  6.14267 
7;+  /  =  I4h  29m  58.8 

log  x"  =  9.76720,, 

This  is  now  the  Greenwich  mean  time  corresponding  to 
the  Washington  sidereal  time  A.  In  order  to  compare  the 
two,  T^-\-  t  must  be  converted  into  sidereal  time. 

T,  +  /  =  Hh  29™    5S.8 

Table  III,  Appendix  N.  A.,  2    22  .77 

Sidereal  time  Greenwich  M.  N.  =    4     4    48  .56 
Greenwich  sidereal  time  00  =  18   36    17.1 

A  =  @0  —  A  =  5h  8m  I2M, 

the^equired  difference  of  longitude. 

238.  If  the  ephemeris  were  perfect,  very  little  could  be 
done  further  in  the  way  of  perfecting  this  method.  The 
errors  of  the  ephemeris,  however,  are  not  inconsiderable,  and 
in  consequence  it  cannot  be  used  directly  as  above,  except 
when  an  approximate  value  of  the  longitude  is  sufficient. 
For  the  year  1877  tne  average  correction  to  the  right  ascen- 


§  239-  LONGITUDE  BY  MOON  CULMINATIONS.  417 

sions  of  the  ephemeris,  as  derived  from  66  observations  at 
Washington,  was  — s.3i,  which  would  have  produced  an  error 
of  8s  in  the  longitude  if  the  observations  had  been  used  for 
that  purpose. 

Either  of  two  different  methods  may  be  used  for  eliminat- 
ing from  the  result  these  errors  of  the  ephemeris. 

First.  Correction  of  the  ephemeris.  This  method  is  due  to 
Prof.  Peirce.*  The  ephemeris  is  compared  with  all  available 
observations  of  the  moon  made  at  Greenwich,  Washington, 
and  other  fixed  observatories  during  the  lunation,  and  in 
this  way  a  series  of  corrections  to  the  ephemeris  obtained 
which,  as  they  depend  on  all  available  data,  are  much  more 
reliable  than  simply  the  place  of  the  moon  observed  at  any 
one  observatory. 

Peirce  found  that  lor  each  semilunation  the  corrections  to 
the  right  ascension  of  the  ephemeris  could  be  represented 
by  the  formula 

X  =  A  +  Bt  +  Cf\ (405) 

X  being  the  correction  required,  t  the  time  reckoned  from 
any  assumed  epoch  (which  should  be  chosen  near  the  mid- 
dle of  the  period  under  consideration  for  greater  conven- 
ience), and  A,  B,  and  C  being  constants  determined  from  the 
observations  made  at  Washington,  Greenwich,  etc.  The 
ephemeris  when  so  corrected  is  used  as  already  explained. 

239.  Second.  Corresponding  observations  The  difference  in 
the  longitude  of  any  two  points  mav  be  found  by  com- 
paring the  values  of  the  right  ascension  of  the  moon  ob- 
served on  the  same  night  at  both  places. 

The  times  of  transit  of  the  moon's  bright  limb  and  of  the 
comparison  stars  are  observed  at  both  places  and  the  cor- 
rections applied  as  already  explained  to  find  the  right  ascen- 

*  Report  of  U.  S.  Coast  Survey  1854,  p.  115  of  Appendix. 


41  8  PRACTICAL   ASTRONOMY.  §  239. 

sion  of  the  centre  at  the  instant  of  transit.  It  will  be  a  lit- 
tle better  if  the  same  comparison  stars  are  used  at  both 
stations. 

Let  L,  and  Za  =  the  assumed  longitudes  of  the   two  sta- 

tions ;  * 

A  =  the  true  difference  of  longitude  ; 
Al  and  A^  =  right   ascensions   of  moon's  centre   from 

observations  at  Ll  and  Z2; 

H  —  variation  of  right  ascension  for  one  hour 
of  longitude,  while  passing  from  meri- 
dian of  Zj  to  that  of  Lz. 

Then  A,  -  A,  =  \H\ 


H  is  taken  from  the  table  of  moon  culminations,  where  it 
is  given  for  the  instant  of  transit  of  the  moon's  centre  over 
the  meridian  of  Washington.  When  used  as  in  (406)  its 
value  must  be  interpolated  for  a  longitude  midway  between 
Ll  and  Z2. 

Example.  As  an  example  of  the  determination  of  longitude  by  corresponding 
observations,  let  us  take  the  transit  of  the  moon,  the  observations  and  reduction 
of  which  are  given  in  Art.  196. 

We  have  there  found  for  1883,  October  15  : 

Right  ascension  of  moon's  first  limb,     ih  I5m  5O*.o8. 
Second  f  limb,     i    18     11.76. 

At  Washington  the  right  ascensions  of  the  limbs  were  observed  as  follows  : 

First  limb,     ih  i6m    78.38. 
Second  limb,     i    18    28  .69. 

*  Reckoned  from  Washington  or  Greenwich  according  as  we  use  the  epheme- 
ris  computed  for  Washington  or  Greenwich.  One  of  the  longitudes,  LI  or  Z», 
must  be  known  with  some  accuracy. 

f  This  is  corrected  for  defective  illumination. 


§239- 


LONGITUDE  BY  MOON  CULMINATIONS. 


419 


Taking  the  mean  in  each  case  as  the  observed  right  ascension  of  the  centre, 
we  have 

A*  =  i    17    18  .035. 
A^—Al  =    I7MI5- 


From  ephemeris,  H  = 


I538.88; 


A.  =  —^-77 — -  =  oh.ni2  =  6m  40s. 3. 
H 

The  difference  of  longitude  between  Washington  and  Bethlehem  determined 
telegraphically  is  6m  40*. 2.  This  close  agreement  is  of  course  accidental ;  a 
deviation  of  four  or  five  seconds  from  the  true  value  would  not  have  been  sur- 
prising. 

If  we  reduce  the  observations  of  the  two  limbs  separately,  we  find  : 

First  limb,     A  =  6m  44'.  7. 
Second  limb,     A.  =  6    36  .o. 

The  mean  being  the  same  as  above.  This  is  an  illustration  of  the  necessity  of 
employing  transits  of  both  limbs.  Frequently  the  difference  of  longitude  de- 
termined separately  from  transits  of  each  limb  will  show  much  wider  deviations 
than  this,  even  when  all  possible  care  is  taken  to  avoid  error. 

To  illustrate  the  method  of  Art.  236  for  deriving  the  moon's  right  ascension 
by  means  of  comparison  stars,  take  the  following  transits  of  the  moon  : 
f  Piscium  and  v  Piscium  observed  at  the  Sayre  observatory,  1883,  October  15. 


Object. 

Clock 

Time. 

f  Piscium 

I1 

nm 

558-67 

Moon    I 

i 

15 

55  -55 

Moon  II 

i 

18 

17  .23 

v  Piscium 

I 

35 

30  .41 

These  times  are  corrected 
for  instrumental  errors,  and 
that  of  the  second  limb  of 
the  moon  for  defective  illu- 
mination. The  clock-rate 
is  inappreciable. 


/"Piscium. 

v  Piscium. 

/Piscium. 

v  Piscium. 

e 

xh  nm  558.67 

i"  35m  3°8-4i 

a 

h  ZIm  50*.  06 

i»  35m  248-87 

e 
©' 

0    -  9 

i    i5     55  -55 
i    18     17.23 
4-    3     59  -88 

1     15     55  -55 
i    18     17  .23 
-  19     34  -86 

A* 
AQ' 
Mean  of  A  0 

15     49  -94 
18     ii  .62 
15     49  .98 

i    15     50  .01 
i    18     ii  .69 

0'-  9 

-f    6    21  .56 

-17     13-18 

-*• 

18     ii  .66 

420  PRACTICAL   ASTRONOMY.  §  240. 

This  method  of  deriving  the  moon's  right  ascension  is  employed  with  most 
advantage  when  the  same  comparison  stars  are  used  at  both  places  whose  dif- 
ference of  longitude  is  required,  as  then  uncertainties  in  the  places  of  the  stars 
will  produce  no  appreciable  effect  on  the  result. 

In  our  example  we  have  preferred  to  use  the  value  of  the  moon's  right 
ascension  derived  in  Art.  196,  since  the  value  of  A  T  there  used  was  obtained 
from  transits  of  a  number  of  stars,  and  thus  a  result  obtained  more  likely  to  be 
reliable  than  the  one  above,  which  depends  only  on  two  stars. 

240.  If  the  difference  in  longitude  between  the  two  places  is  more  than  two 
hours,  the  above  method  requires  some  modification,  as  then  the  third  differ- 
ences in  the  hourly  motion  H  will  be  appreciable. 

The  right  ascensions  AI  and  A*  are  obtained  from  observation  precisely  as 
before  ;  then  the  right  ascensions  are  taken  from  the  ephemeris  for  the  time  of 
culmination  at  the  two  meridians,  using  for  this  purpose  the  assumed  values  of 
the  longitude. 

Let  a.1  and  aa  =  values  of  the  right  ascension  taken  from  the  ephemeris  for 

the  assumed  longitudes  LI  and  Za  ; 
Aa  =  correction  to  the  ephemeris. 

Then         <*!  -f-  Aa  and  a*  -f-  Aa  =  true  values  of  the  right  ascension. 

If  then  Za  and  LI  are  the  true  values  of  the  longitude,  («a  -f-  ^a)  —  (<*i+  Act) 
=  a?  —  at  will  be  equal  to  Az  —  AI. 

Let  Li  —  Li  -f-  AL  =  true  difference  of  longitude.  Then  AL  is  the  correc- 
tion to  the  assumed  difference  of  longitude. 

Let  K  =  (At  —  AI)  —  (a*  —  «i). 

Then  AL~J{  .......  "...     (407) 

H  being,  as  above,  the  hourly  change  in  the  moon's  right  ascension,  AL  will 
here  be  expressed  in  hours.     To  reduce  to  seconds  we  multiply  by  3600,  viz., 


(4oS) 


This  process  is  sufficiently  simple  in  theory,  but  if  the  table  of  moon  culmina 
tions  is  employed  the  moon's  right  ascension  must  be  interpolated  to  fourth  or 
fifth  differences,  which  will  involve  considerable  labor.  By  using  the  hourly 
ephemeris  of  the  moon  the  interpolation  need  only  be  carried  to  second  differ- 
ences. In  any  case  we  must  assume  the  moon's  motion  in  right  ascension 
given  in  the  ephemeris  to  be  correct. 


§241.  LONGITUDE  BY  MOON  CULMINATIONS.  421 

The  hourly  motion,  //",  is  taken  from  the  ephemeris  for  the  time  of  observa- 
tion at  the  meridian  whose  longitude  is  to  be  determined. 

Example.  1883,  October  16,  the  moon's  right  ascension  was  determined  by 
meridian  observation  at  Greenwich  and  Bethlehem  as  given  below.  The 
transit  of  the  second  limb  was  observed,  the  Bethlehem  observations  being 
made  and  reduced  precisely  as  in  the  example  of  Art.  196. 

At  Greenwich,     AI  =  2h    6m  17".  46. 
At  Bethlehem,     A*  =  2    19    32  .18. 

From  the  houriy  ephemeris  of  the  moon  we  now  take  the  right  ascension  of 
the  moon's  centre.  Since  the  argument  is  the  Greenwich  mean  time,  we  must 
'convert  the  above  values  of  the  right  ascension,  which  are  equal  to  the  sidereal 
times  of  observation,  into  the  corresponding  Greenwich  mean  solar  time, 
using  for  the  longitude  of  Bethlehem  the  best  approximation  to  the  true  value 
which  we  possess.  Thus  : 

Local  sidereal  time  ...............  A*  =  2h  19™  32M8 

Assumed  longitude  from  Greenwich.  5  i  31  .9 

Greenwich  sidereal  time  ...........  2h    6m  17s.  46  7  21  4.08 

Sidereal  time,  mean  noon  ..........  13    38  38.61  13  38  38.61 

Sidereal  interval  past  noon  ........  12    27  38.85  17  42  25.47 

Table  II,  Appendix  of  Ephemeris.  .  2  2  .48  2  54  .05 

Greenwich  mean  time  .............  12    25  36.37  17  39  31.42 

For  these  times  we  find  ..........     ax  =  2h    6m  I7s.6i         aa  =  2h  19™  32S.38 

From  the  table  of  moon  culminations  —  page  379  of  Ephemeris  —  we  find  for 
the  hourly  motion  in  right  ascension  at  the  time  of  the  Bethlehem  observation, 


Then  by  formula  (408),     AL  =  —  ".05  X  -~ 

150-50 

We  have  assumed  L  = 

Final  value  of  longitude,  5    i    30  .8.  t 

241.  The  determination  of  the  moon's  right  ascension  by  the  difference  be- 
tween the  time  of  transit  of  the  moon  and  a  neighboring  star  does  not  do  away 
with  the  necessity  for  correcting  the  observed  times  for  all  known  errors  of  the 
transit  instrument  as  explained  in  Articles  195  and  196.  What  we  require  is 
the  right  ascension  of  the  moon's  centre  at  the  instant  of  transit  over  the 
meridian  of  the  place  of  observation.  Since  this  right  ascension  is  constantly 
changing,  if  there  is  an  uncorrected  error  of  T  seconds  in  the  reduced  time,  it 


422  PRACTICAL   ASTRONOMY.  §  241. 

is  precisely  the  same  as  though  the  moon  were  observed  with  an  instrument 
perfectly  mounted  in  a  meridian  differing  from  this  one  by  r  seconds.  Thus 
an  uncorrected  instrumental  error  affects  the  resulting  longitude  by  its  full 
amount. 

In  order  to  obtain  the  best  result  from  the  method  of  moon  culminations  the 
observations  should  be  arranged  so  as  to  include  about  an  equal  number  of  each 
limb  ;  that  is,  the  moon  should  be  observed  about  the  same  number  of  times 
before  and  after  full  moon.  In  this  way  uncertainties  in  the  value  of  the  semi- 
diameter  will  be  eliminated,  and  to  some  extent  the  personal  equation  of  the 
observer  in  estimating  the  instant  of  transit  of  the  limb.  As  the  difference 
between  the  values  of  the  longitude,  determined  from  the  first  and  second  limbs 
respectively,  from  observations  embracing  an  entire  year,  frequently  amounts  to 
io8,  the  importance  of  this  will  be  obvious. 

In  a  discussion  of  the  limit  of  accuracy  attainable  in  the  determination  of 
longitude  by  moon  culminations,  Prof.  Peirce  gives*8. 101  as  the  probable  error 
of  a  single  determination  of  the  right  ascension  of  the  moon.  The  probable 
error  of  the  difference  between  two  observed  right  ascensions  would  then  be 
•.142  ;  the  probable  error  of  the  resulting  longitude  is  twenty-seven  times  this, 
or  38.83.  By  using  an  ephemeris  corrected  as  before  explained,  this  probable 
error  of  a  single  determination  is  somewhat  reduced. 

If  now  the  law  of  distribution  of  error,  which  forms  the  basis  of  the  method 
of  least  squares,  were  the  only  thing  to  be  considered  in  making  and  combining 
observations,  we  could  by  a  sufficient  accumulation  of  individual  determinations 
reduce  this  probable  error  to  an  unlimited  extent.  In  this  case,  however,  as  in 
all  cases  where  quantities  are  determined  by  observation,  the  errors  of  a  purely 
accidental  character  are  so  combined  with  others  of  a  constant  character  that 
accumulation  of  observation  beyond  a  certain  limit  adds  but  little  to  the  accu- 
racy of  the  final  result. 

Prof.  Peirce  estimates  the  ultimate  limit  of  accuracy  which  we  can  hope  to 
reach  in  determining  a  quantity  by  observation  at  about  four  times  the  accuracy 
of  the  most  carefully  executed  single  determination.  If  then  we  assume  that  it 
is  possible  to  determine  the  difference  in  the  moon's  right  ascension  within  8. 1 
by  a  single  observed  transit  at  each  place,  this  would  give  a  value  of  the  longi- 
tude accurate  to  within  28.7.  The  ultimate  degree  of  accuracy  which  could  be 
attained  would  then  be  within  ".67  of  the  truth.  Owing,  however,  to  the  unex- 
plained discrepancies  in  the  results  from  the  two  limbs  of  the  moon,  this  ulti- 
mate error  is  probably  too  small.  Prof.  Peirce  places  the  limit  at  is.oo,  a  limit 
which  might  be  reached  by  observing  all  available  culminations  for  two  or  three 
years,  but  which  would  not  be  much  reduced  by  a  further  accumulation  of 
observations. 

*  Report  of  U.  S.  Coast  Survey  1854,  p.  112  of  Appendix. 


§  243-        LONGITUDE  BY  OCCULTATIONS  OF  STARS.  423 


Determination  of  Longitude  by  Occult *  at  ions  of  Stars. 

242.  The  observation  of  occultations  of  stars  by  the  moon 
and  of  eclipses  of  the  sun  furnishes,  next  to  the  telegraphic 
method,  the  most  accurate  means  of  determining  the  differ- 
ence of  longitude  between  two  places.*     Prof.  Peirce  esti- 
mates the  ultimate  accuracy  attainable  by  this  method  as 
within  one  tenth  of  a  second  of  time. 

The  mathematical  theory  of  eclipses  and  occultations  of 
stars  and  of  planets  by  the  moon,  and  of  fixed  stars  by 
planets,  may  all  be  embraced  in  one  general  discussion.  It 
is  not  proposed  to  enter  here  into  the  general  problem  of 
eclipse  prediction,  as  it  would  lead  us  beyond  what  is  de- 
signed to  be  the  scope  of  this  work.  We  shall  therefore  con- 
fine ourselves  to  so  much  of  the  problem  as  relates  to  the  oc- 
cultation  of  fixed  stars  by  the  moon. 

General  Theory. 

243.  The  distance  of  a  fixed  star  is  so  great  in  comparison 
with  the  distance  of  the  moon  that  the  rays  of  light  from 
the  star  enveloping  the  moon  may  be  regarded  as  forming  a 
cylindrical  surface,  the  radius  of  the  cylinder  being  equal  to 
the  radius  of  the  moon.    If  this  cylinder  intersects  the  earth, 
the  star  will  be  hidden  from  all  parts  of  the  earth's  surface 
within  the  cylinder.     Let  a  line  be  supposed  drawn  from  the 
star  through  the  centre  of  the  moon  :  this  line  will  form  the 
axis  of  the  cylinder,  and  the  point  where  it  pierces  the  celes- 
tial sphere  coincides  with  the  place  of  the  star. 

*  When  the  places  are  favorably  situated  for  a  chronometric  determination 
that  method  may  be  preferable,  but  a  high  degree  of  precision  is  not  possible 
when  the  chronometers  are  transported  by  land. 


424 


PR  A  C  7  '1C A  L   AS  TRONOM  Y. 


§244- 


Now  let  a  plane  be  passed  through  the  centre  of  the  earth 
perpendicular  to  this  line:  this  plane  is  called  the  funda- 
mental plane,  and  is  taken  as  the  plane  of  XY  m  considering 
the  rectangular  co-ordinates  of  the  points  entering  into  the 
problem.  The  axis  of  X  is  the  line  in  which  the  funda- 
mental plane  intersects  the  plane  of  the  equator,  the  positive 
axis  of  Fis  directed  towards  the  north,  and  the  axis  of  Z  is 
parallel  to  the  axis  of  the  cylinder;  the  origin  of  co-ordinates 
being  the  centre  of  the  earth. 

•  244.    To  find  the  distance  of  any  point  on  the  earth's  surface 
from  the  axis  of  the  cylinder. 

Let  a,  d  =  the  right  ascension  and  declination  of  the  star; 

A,D,  r  =  the  right  ascension,  declination,  and  distance 

from  the  centre  of  the  earth,  of  the  moon's 

centre  ; 

x,  y,  2  —  \\-\Q    rectangular    co-ordinates   of   the   moon's 

centre. 

Let  the  axis  of  X  be  positive  in  the  direction  of  the  end 
whose  right  ascension  is  equal  to 
90°  +  a. 

Then  E,  Fig.  49,  being  the  centre 
of  the  earth,  M  the  moon,  and  P  the 
pole,  we  have 

x  =  r  cos  MX', 
y  =  r  cos  MY\ 
z  =  r  cos  MZ. 


Y      From  the  triangle  MPX% 
PX  =  90°;        MPX  =  90°  -  (A  -  a}. 


FIG.  49. 

MP  =  90°  -  D ; 

Therefore  cos  MX  —  cos  D  sin  (A  —  a). 

Similarly  from  triangles  MPZ  and  MPY  we  find  the  values 


§  245-        LONGITUDE  BY  OCCULTATIONS  OF  STARS.  425 

of  cos  MZ  and  cos  MY,  from  which  result  the  following 
equations  : 

x  =  r  cos  D  sin  (A  —  a)\  \ 

y  —  r[sin  D  cos  3  —  cos  D  sin  d  cos  (A  —  «)];  I  (409) 

s  =  r[sin  D  sin  tf  +  cos  D  cos  tf  cos  (A  —  or)].  ) 

As  the  axis  of  the  cylinder  is  parallel  to  the  axis  of  Z  and 
passes  through  the  centre  of  the  moon,  x  and  y  will  be  the 
co-ordinates  of  the  point  where  this  axis  pierces  the  funda- 
mental plane. 

For  our  purposes  z  will  not  be  required.  For  computing 
x  and  y  with  extreme  accuracy  it  is  convenient  to  transform 
(409)  as  follows: 

Let  TT  =  the  equatorial  horizontal  parallax  of  the  moon. 
Then    r  —  -  -  ,  expressed  in  terms  of  the  equatorial  radius 
of  the  earth, 

cos  D  sin  (A  —  a) 
and  x  —   ---  .—  —  ; 

sin  n 

sin  (D-3)cos^(A-<x)+s'm(D+d)sm^(A-a)  , 
y  —  ~  —  =  —  -  -  .(410) 

sin  n 


245.  Let  £,  77,  and  <2  =  the  rectangular  co-ordinates  of  a  point 

on  the  earth's  surface  ; 

p  =  the  line  joining  this  point  with  the 
centre  of  the  earth  ; 

q>  =  the  geographical  latitude  of  this 
point  ; 

cp'  =  the  geocentric  latitude; 

*  =  the  local  sidereal  time. 


PRACTICAL  ASTRONOMY.  §  245- 

Then  in  Fig.  49,  if  we  suppose  M  to  be  a  point  on  the  sur- 
face of  the  earth  whose  co-ordinates  are  £,  77,  and  8»  we  have 


£  =  p  cos  JO";        77  =  p  cos  MY;        2  =  p  cos 
In  the  triangle  MPX 

MP  =  90°  -  q>'\         MPX  =  90°  -  O  -  «);          /^  =  90°. 
Therefore          cos  MX  —  cos  q>'  sin  (//  —  a). 
In  the  triangle  J//>F 

PY  =  6;        MPY  =  180°  -  O  -  a). 
Therefore 

cos  J/F  =  sin  <p'  cos  #  —  cos  <pf  sin  (J  cos  (/^  —  a), 
and  similarly  for  cos  J/Z,  so  that  finally 

g  =  p  cos  <??'  sin  (/w  —  a)  ;  j 

T;  =  p[sin  (f)'  cos  tf  —  cos  <pf  sin  (J  cos  (/<  —  a)]  ;  v  (41  1) 

<?  =  p[sin  ^'  sin  tf  -f-  cos  (p'  cos  d  cos  (j*  —  or)].  ) 

These  formulae  may  be  computed  in  this  form,  or  they  may 
be  adapted  to  logarithmic  computation,  as  follows  : 


p  sin  tp'  =  b  sin  B; 

p  cos  (pf  cos  (p  —  a)  =  ^  cos  ^ 

g  =  p  cos  <£/  sin  (yw  —  or)  ; 
?;  ~  ^  sin  (^  —  d) ; 
2  =  b  cos(B  —  d). 


(412) 


(//  —  or)  is  the  hour-angle  of  the  star  as  seen  from  the  given 
point  on  the  earth's  surface  at  the  instant  for  which  £,  77, 
and  3  are  computed. 


§246.        LONGITUDE  BY  OCCULTATIONS  OF  STARS.  427 

Let  H  —  the  Washington  hour-angle  of  the  star ; 
h^  =  the  local  hour-angle  =  ^  —  a- 
A  =  the  west  longitude  of  the  point  £,  T;,  £. 

Then  h0  =  H  -  A (413) 

Let  A  —  the  distance  of  the  point  from  the  axis  of  the 
shadow. 


Then  A  =  \/(x  -  £)3  +  (y  -  rff (414) 

At  the  instant  of  the  beginning  or  ending  of  an  occulta- 
tion,  it  is  evident  that  the  point  £,  v>  £  will  be  in  the  surface 
of  the  cylinder,  and  the  distance  from  the  centre  A  is  equal 
to  the  radius  of  the  cylinder,  which  in  turn  is  equal  to  the 
radius  of  the  moon,  or  .2723,  expressed  in  terms  of  the  earth's 
equatorial  radius.  Therefore 

The  condition  for  the  beginning  or  ending  of  an  occupation  at 
any  place  is 


k  =  .2723  =  V(*  -  *)•  +  (j>-  '/Y-   •    •   (415) 


Prediction  of  the  Principal  Phases  of  an  Occultation.     . 

246.  The  instant  of  beginning  and  ending  of  an  occultation 
are  called  respectively  the  time  of  immersion  and  emersion. 
We  shall  at  first  suppose  it  to  be  known  that  an  occultation 
of  the  star  under  consideration  will  be  visible  from  the  given 
place  on  a  given  day,  and  shall  develop  the  formulas  for  de- 
termining these  two  phases — viz.,  of  immersion  and  emersion. 

For  this  purpose  we  require  the  solution  of  equation  (415) 
for  the  local  time  T,  of  which  x,  y,  5,  and  /;  are  functions- 
The  equation  is  transcendental  and  of  such  a  form  that  a 
direct  solution  is  not  possible.  In  fact  it  will  readily  appear 
that  an  infinite  number  of  values  of  T  must  satisfy  this  equa- 


428  PRACTICAL  ASTRONOMY.  §  247. 

tion,  since  the  same  star  may  suffer  occultation  an  indefinite 
number  of  times. 

Equation  (415)  must  therefore  be  solved  by  approximation, 
the  most  convenient  method  being  as  follows :  x  and  y  are 
computed  for  a  time  T^as  near  as  may  be  to  that  of  the  re- 
quired phase.  For  the  first  approximation  the  time  chosen 
is  commonly  that  of  the  geocentric  conjunction  of  the  moon 
and  star  in  right  ascension.  This  time  is  readily  found  from 
the  hourly  ephemeris  of  the  moon  by  finding  the  Green- 
wich time  when  the  moon's  right  ascension  is  equal  to  the 
right  ascension  of  the  star.  If,  as  will  commonly  be  the  case 
in  the  United  States,  the  meridian  from  which  the  longitude 
is  reckoned  is  that  of  Washington,  the  above  time  will  be 
converted  into  Washington  time  by  subtracting  the  differ- 
ence of  longitude  between  Washington  and  Greenwich,  viz., 
5h  8m  I2S.09. 

The  object  of  this  computation  will  generally  be  to  deter- 
mine the  time  of  immersion  and  emersion,  to  assist  in  ob- 
serving the  occultation.  For  this  purpose  great  accuracy  will 
not  be  necessary ;  in  fact  an  error  of  a  whole  minute  in  the 
computed  time  would  not,  ordinarily,  be  a  serious  matter. 
The  general  formulas  may  therefore  be  much  abridged.  In 
any  case  it  would  be  superfluous  to  use  the  rigorous  formulas 
in  the  first  approximation. 

247.  We  first  compute  x,  y,  £,  and  77  for  the  instant  of 
geocentric  conjunction  of  the  moon  and  star  in  right  ascen- 
sion, viz.,  when  A  =  a.  For  this  instant  (410)  may  be  writ- 
ten 

D-d 

x  =  o;        y  =  — ; (416) 

sin  n 

For  the  short  interval  between  conjunction  and  immersion 
or  emersion  we  may  then  assume  the  change  in  x  and  y  to 
be  proportional  to  the  time. 


§  248.       LONGITUDE  BY  OCCULTATIONS  OF  STARS.  429 

Let  x'  and  y'  =  the  changes  in  x  and  y  in  one  hour,  mean 
solar  time. 

Differentiating  the  expression  for  x  in  (410),  and  for  y  in 
(416),  we  have  for  the  instant  of  conjunction 

dA  dD 

dx  =  -s  --  cos//;          ay  = 


sin  TT  sin  TT 


Let  AA  and  AD  =  the  hourly  changes  in  the  moon's  right 
ascension  and  declination  taken  from 
the  ephemeris. 

AA  AD 

Then  *'  =          cos^         -:s-"''    •    •    (4I7) 


x,  y,  ;r'and  y,  being  independent  of  the  place  of  observation, 
may  be  computed  for  any  future  time,  and  will  be  available 
for  all  parts  of  the  earth  from  which  the  occultation  is  visi- 
ble. Their  values  are  given  in  the  American  Ephemeris 
for  all  the  principal  stars  occulted  throughout  the  year. 
When  required  for  this  purpose  they  may  therefore  be  taken 
directly  from  that  publication. 

248.  We  must  next  compute  £,  77,  £'  and  77'  —  the  latter  be- 
ing the  change  in  £  and  77  for  one  hour  mean  solar  time. 
B,  and  77  are  given  by  formulas  (411)  or  (412). 

For  computing  £'  and  77'  we  differentiate  the  first  and  sec- 
ond of  (411)  with  respect  to  (//  —  or),  viz.: 

d%  =  p  cos  cp'  cos  (fit  —  a)  {/(fit  —  or); 

drj  =  p  cos  q>'  sin  d  sin  (/w  —  a)  d(^L  —  a). 


(p  —  a)  is  the  hour-angle  of  the  star.  Let  us  now  substitute 
for  d(n  —  a)  the  change  which  takes  place  in  the  value  of 
this  hour-anele  in  one  mean  solar  hour. 


•a 

_    Th 


ihomos  mean  solar  time  —  Ihom9s.856  sidereal  time  =  54148". 


43°  PRACTICAL  ASTRONOMY.  §  248. 

Therefore  d(p  —  a)  =  54148"  sin  i" (418) 

2  =  [9.4I9I57]  P  cos  <p'  cos  (/£  -  a);  )          , 

rf  =  [9.419157]  p  cos  <p'  sin  (/*  —  a)  sin  d.  )  ' 

Let  k  —  the  moon's  radius  expressed  in  terms  of  the  earth's 

radius  =  .2723 ; 

T  —  approximate  time  of  immersion  or  emersion;* 
T  -\-  r  =  true  time  of  phase. 

T  will  then  be  an  unknown  correction  to  Tto  be  determined. 
•*"»  y>  £»  and  ?;  having  been  computed  for  the  time  J",  their 
true  values  will  be 

x  +  X'T,        y  +  y  r;         £  +  Z'r;         ??  +  ^r.      (420) 

Let  the  auxiliary  quantities  Q,  m,  M,  n,  N  be  determined 
as  follows : 

sinG  =  (*-£)  +  (*'-£')';! 


m  sin  M  —  (x  —  £);         n  sin  7^  =  (x1  —  £')  ;  )      /       , 
m  cos  M  =  (  y  —  17);        n  cos  N  =  (y1  —  rf\  \  *  V4 

Then  (421)  become 

k  sin  Q  •=  m  sin  J/  -f-  r«  sin  A'; 
k  cos  Q  =  m  cos  J/  -f-  T;Z  cos  A7". 

From  these  we  derive 

k  sin  (Q  -  N)  =  m  sin  (J/  -  JV); 

^  cos  (Q  —  N)  =m  cos  (J/  —  N  )  +  «r. 


*  For  the  first  approximation  the  time  of  conjunction  in  right  ascension  may 
be  used  as  before  explained. 

f  It  will  be  observed  that  these  two  equations  are  identical  with  (415). 


§  249-       LONGITUDE  BY  OCCULTATIONS  OF  STARS.  431 


Let  us  write  Q  —  N  =  $. 

m  sin  (M  —  N} 
Then  sin  ^  =  —, '; 


_  k  cos  $       m  cos  (M  —  N) 


(423) 


Thus  we  have  our  equation  solved  for  f  and  consequently 
for  T  -\-  T.  Since  the  algebraic  sign  of  cos  $  is  not  deter- 
mined,  the  last  equation  gives  two  values  of  r,  that  value  cor- 
responding to  the  minus  sign  of  cos  $>  giving  the  time  of 
immersion,  that  given  by  the  plus  sign  being  the  time  of 
emersion. 

The  resulting  times  will  only  be  approximations  to  the 
true  values,  since  in  deriving  them  we  have  neglected  the 
second  and  higher  orders  of  differences  in  the  variation  of 
x,  y,  £,  and  17. 

If  we  require  the  time  more  accurately,  we  may  now  as- 
sume these  approximate  values  of  7"and  recompute  formulae 
(41 1),  (419),  (422),  and  (423),  thus  obtaining  a  second  approxi- 
mation to  the  values  of  T  for  immersion  and  emersion. 

Position  A  ngle  of  the  Star. 

249.  The  accurate  observation  of  the  star's  emersion  will 
be  greatly  facilitated  if  we  know  in  advance  the  exact  point 
on  the  moon's  limb  where  its  appearance  may  be  expected. 
This  point  is  determined  by  its  position  angle,  which  is  the 
angle  measured  from  the  north  point  of  the  moon's  limb 
around  towards  the  east  to  the  point  in  question.  We  may 
perhaps  define  this  angle  more  clearly  as  follows : 

Suppose  two  great  circles  drawn  from  the  moon's  centre 
respectively  through  the  pole  and  the  star:  the  position  angle 
will  then  be  the  angle  between  these  circles,  measured  from 
that  drawn  through  the  pole  around  towards  the  east. 


43-2 


PR  A.  C  TIC  A  L   AS  TR  ONOM  Y. 


§249- 


In  equations  (421)  x,y,£,  and  ?;  being  the  rectangular  co- 
ordinates of  the  moon's  centre,  and  of  the  place  of  observa- 
tion on  the  earth's  surface,  let  us  suppose  a  system  of  rect- 
angular axes  drawn  through  the  latter  point  and  parallel  to 
the  old  axes,  (x  —  £)  and  (y  —  //)  will  be  the  rectangular 
co-ordinates  of  the  moon's  centre  in  reference  to  this  new 
system. 

Since  k  is  the  moon's  radius,  equations  (421)  require  Q  to 
be  the  position  angle  of  the  moon's  centre,  measured  from 
the  axis  of  Y»  Now  it  is  evident  that  when  the  star  is  in 
contact  with  the  moon's  limb,  which  is  the  condition  ex- 
pressed by  equations  (421),  the  position  angle  of  the  star 
measured  from  the  north  point  of  the  moon's  limb  will  differ 
from  the  position  angle  of  the  moon's  centre  measured  from 
the  axis  of  Fby  180°.  - 


Thus,  in  Fig.  50,  the  star  at  immersion  being  atA,JVMA 
is  the  position  angle  required.  Calling  this  angle  P,  we 
have 

P=  Q  -  180°. 


§250.        LONGITUDE  BY  OCCULTATIONS  OF  STARS.  433 

At  emersion,  as  shown  in  Fig.  51,  the  position  angle  Pwill 
be  the  angle  NES  WA .     Therefore 


P=  0-f 
Then  since,  equations  (423),  Q  =  N  -\-  ip,  we  have 

For  immersion  P  =  N  +  fy  —  180°;  )  ,      ^ 

For  emersion     P  =  N  +  $  +  180°.  | 

If  the  telescope  used  is  mounted  equatorially  and  provided 
with  a  position  micrometer,*  this  point  may  be  kept  in  view 
very  readily  by  placing  the  micrometer-thread  tangent  to 
the  moon's  limb  at  the  point. 

If  the  telescope  is  not  provided  with  a  micrometer,  a  sin- 
gle  thread  may  be  placed  in  the  focus  of  a  common  eye- 
piece, and  a  rough  graduation  marked  around  the  rim.  This 
thread  may  then  be  set  in  the  direction  of  the  tangent  to  the 
moon's  limb  as  before. 

250.  Ii  the  telescope  has  only  an  altitude  and  azimuth  mo- 
tion, it  will  be  convenient  to  measure  the  angle  from  the  ver- 
tex, or  highest  point  of  the  moon's  limb,  instead  of  the  north 
point. 

Consider  the  triangle  formed  by  the  zenith, 
the  pole,  and  the  moon's  centre. 

Let  V  —  the  position  angle  measured  from 
the  moon's  vertex. 

Then,  referring  to  Fig.  52, 

/ 

V  =  P  -  C. 


*  In  a  position  micrometer  the  reticule  revolves  in  a  plane  perpendicular  to 
the  line  of  colhmation  of  the  telescope,  and  the  threads  may  be  placed  at  any 
angle  with  the  meridian  by  means  of  a  graduated  circle.  On  the  other  hand, 
by  the  same  circle  the  angle  formed  with  the  hour-circle  of  a  star  by  the  line 
joining  it  with  any  other  star  in  the  field  of  the  telescope  may  be  measured. 


434  PRACTICAL   ASTRONOMY.  §2$ I. 

To  determine  C,  apply  to  the  triangle  the  formulae  of 
spherical  trigonometry,  viz.: 

sin  Z  sin  C  =  cos  q>  sin  (/*  —  A)\  \  ,    ^. 

sin  Z  cos  (7  =„  sin  cp  cos  tf  —  cos  (p  sin  tf  cos  (n—A).  \ 

Since  C  will  not  be  required  with  extreme  precision,  and 
at  the  time  for  which  C  is  required  the  right  ascension  of 
the  star  differs  but  little  from  that  of  the  moon,  we  may 
write,  bearing  in  mind  the  values  given  by  equations  (411), 

sin  Z  sin  C  =  £;}  ,      , 

sin  ZcosC  =?;)' 

and  since  at  the  instant  of  contact  the  values  of  £  and  tj  are, 
by  equations  (420),  4;  +  £'r  and  TJ  +  rfr, 


251.  In  connection  with  the  elements  for  predicting  the 
occultation  of  a  given  star,  found  in  the  American  Ephemeris, 
there  are  given  the  limiting  parallels  of  latitude  within 
which  the  star  will  be  occulted.  It  does  not  necessarily  fol- 
low, however,  that  because  a  place  is  within  the  limits  there 
given  the  star  will  be  occulted  at  that  place.  The  limiting 
curves  do  not  coincide  with  parallels  of  latitude,  as  we  might 
show  by  investigating  the  theory  farther,  or  as  may  be  seen 
by  referring  to  the  charts  of  solar  eclipses  to  be  found  in  any 
number  of  the  ephemeris. 

In  case  the  point  falls  outside  the  limit  of  occultation,  it 
will  be  shown  in  computing  r  from  equations  (423),  when 
we  should  find  m  sin  (M  —  N)  >  k,  thus  making  sin  ty  >  I,  an 
impossible  value. 

As  the  observation  of  occultations  near  this  limit  is  not  of 


§252. 


PREDICTION   OF  AN  OCCULTATION. 


435 


great  value  for  the  determination  of  longitude,  it  will  not  be 
worth  while  to  make  a  very  close  computation  to  ascertain 
whether  the  occultation  actually  does  occur  when  it  is  found 
to  be  near  the  limit. 

252.  The  successive  steps  in  preparing  to  observe  the  oc- 
cultation of  a  Nautical  Almanac  star  at  a  given  place,  assum- 
ing it  to  be  visible  at  that  place,*  are  therefore  as  follows : 

I.  We  take  from  the  "Elements  for  the  Prediction  of  Oc- 
cultations"  of  the  American  Ephemeris  the  Washington  mean 
time  of  geocentric  conjunction    T0,  the  Washington   hour- 
angle  H,  also  F,  x ',y,  and  the  star's  apparent  declination  8. 

II.  T0  and  H  are  reduced  to  the  local  time  and  hour, 
angle  by  applying  the  correction  for  longitude,  A. 

p  sin  g>f  and  p  cos  <p'  are  to  be  found  by  the  use  of  table  A. 
TABLE  A. 


0 

logF 

logC 

0° 

.00000 

.00291 

5 

.00001 

.00290 

10 

.00004 

.00286 

15 

.00010 

.00281 

20 

.00017 

.00274 

25 

.00026 

.00265 

30 

.00036 

.00255 

35 

.00048 

.00243 

40 

.00060 

.00231 

45 

.00073 

.00218 

50 

.00085 

.00206 

55 

.00098 

.00193 

60 

.00109 

.00182 

65 

.00119 

.00171 

,70 

.00128 

.00163 

75 

.00136 

.00155 

80 

.00141 

.00150 

85 

.00143 

.00147 

90 

.00145 

.00145 

This  table  is  for  computing  p  cos  <p' 
and  p  sin  <p',  which  will  be  given  by 
the  formulae 

p  cos  <p  =  F  cos  <p\ 


p  sin  <p  = 


sin  <p 


*  We  shall  subsequently  show  how  to  select  from  the  list  of  stars  of  the 
American  Ephemeris  those  whose  occultation  is  likely  to  be  visible  from  a 
given  place  on  a  given  day. 


43 6  PRACTICAL   ASTRONOMY.  §  252. 

III.  We  then  compute  £,  ?/,  £',  and  //'  for  the  local  mean 
solar  time,  (T0  —  A),  by  the  formulae 

£  =  pcos  <p'sin//0;  |       ,       . 

?;  =  p  sin  <£/  cos  3  —  p  cos  9'  sin  tf  cos  h^  \ ' 

&>'  =  [9.4192]  p  cos  (p'  cos  //0; 

T/  —  [9.4192]  p  cos  q>'  sin  //0  sin  d 

In  which  /^0  =  H  —  A  =  /<  —  a. 

IV.  w,  J/,  n,  and  ^Vare  computed  by 

m  cos  M  —  y  —  rj\  n  cos  N  =  y'  —  ^;  ) 

w  sin  (M  —  N)  I 

then  ^  and  r  by   sin  ^  =   -          —;--      — -; 
*  k 


&cosif>      m  cos  (M-N) 

~~  ±  ~~  ~~~~~* 


Calling  the  value  corresponding  to  the  plus  sign  rlf  and 
that  corresponding  to  the  minus  sign  r2,  we  have 

Time  of  immersion  =  T0  +  rl  =  Tj 
Time  of  emersion     =  T0  -\-  ra  =  T9. 

V.  With  these  values  T^and  T^  we  now  repeat  the  com. 
putation  for  a  second  approximation  to  the  true  values  of  the 
time  of  immersion  and  emersion.  hQ  in  (411)  and  (419)  will 
become  (hQ  +  rt)  for  immersion,  and  (hQ  +  T2)  f°r  emersion. 
T,  will  give  us  two  values  of  r;  one  a  small  value  giving  a 
more  accurate  time  for  immersion,  the  other  a  large  value 
giving  an  inaccurate  time  of  emersion.  In  the  same  way  T9 


§  253-  PREDICTION  OF  AN  OCCULTATION.  437 

gives  a  small  and  accurate  value  of  r  for  emersion  and  a 
large  inaccurate  value  for  immersion. 

The  values  of  x  and  y  to  be  used  in  this  second  approxima- 
tion will  be  given  by  the  formulae 

x  =  X'TV        y—   Y-\-y'r^         for  immersion, 
and       x  —  x'r^        y—  Y -\- y'rv         for  emersion. 

The  values  of  r  given  above  will  be  expressed  in  hours. 
If  it  is  considered  desirable  to  express  them  in  minutes  we 
'may  use,  instead  of  «,  a  quantity  n ' ,  viz., 

»'  =  So  =  [8-22i8]». 

As  a  check  upon  the  values  of  the  times  finally  obtained, 
we  compute  for  these  times  the  values  of  x,  y,  £,  and  rj.  If 
the  times  are  correct  these  quantities  will  satisfy  the  equation 

.    (x  -  £)*  +  (y  -  rff  =  0.07413. 

253.  Instead  of  carrying  through  the  computation  of  num- 
bers III  and  IV  with  the  hour-angle  h0  of  geocentric  con- 
junction, we  may  obtain  a  rough  approximation  to  the  time 
of  immersion  and  emersion,  as  follows: 

We  first  require  the  interval  of  time  between  geocentric 
and  apparent  conjunction  in  right  ascension.  At  the  instant 
of  apparent  conjunction  x  =  £;  or  writing  for  x  and  £  their 
values, 

r^x'  —  p  cos  cpf  sin  (//0  -|-  r0). 

In  which  r0  is  the  interval  required  and  h0  is,  as  before,  the 
hour-angle  at  the  station  at  the  time  of  geocentric  conjunc- 
tion. 


438  PRACTICAL  ASTRONOMY.  §  253. 

We  have 

sin  (h0  +  ro)  —  si*1  ^o  cos  ro  +  cos  ^o  sm  To 

=  sin  ^0(i  —  2  sin2^r0)  -|-  cos  /i0  2  sin  £r0  cos  £r0; 


and  finally, 

sin  (h0  +  r0)  =  sin  h.  +  2  sin  ^r0  cos  (>&0  +  Jr0). 
r  will  never  be  very  large,  so  we  may  write 
2  sin  £r0  =  r0  54I4877.  sin  i/r  =  [9.4192]  rof 


since  the  unit  in  which  r  is  expressed  is  the  mean  solar  hour. 
Therefore 

rX  =  p  cos  <p'  sin  ^0  +  [9.4192]  p  cos  9'  cos  (^0+ir0)  .  r0. 

Write  p  cos  <p'  sin  //0  —  ^"0;  )  ,      . 

[9.4192]  p  cos  ^  cos  (  ' 


Then  r0  ^  -.-  .......     (430) 

In  the  first  approximation  the  r0  in  the  value  of  £'  may  be 
neglected  ;  or  we  may  assume  it  equal  to  -J^0,  which  will  gen- 
erally be  a  little  more  accurate. 

As  the  average  duration  of  an  occultation  is  about  one 
hour,  we  may  therefore,  in  ordinary  cases,  assume  as  the 
hour-angle  in  equations  (411)  and  (419)  — 

For  immersion,  h,  +  r0  —  3om;  )  /      N 

For  emersion,     h,  +  r0  -f  3om.  )  ' 

The  value  of  r0  may  be  taken  from  Downes's  table,  given 
in  connection  with  the  subject  of  occultations  in  the  Ameri- 
can Ephemeris. 


§253-  PREDICTION  OF  AN  OCCULTATION.  439 

Example. 

Required  the  time  of  immersion  and  emersion  of  the  star  dl  Libra  at  Beth* 
lehem,  1883,  September  6th.         cp  =  40°  36'  24";         A  =  —  oh  6m  4O'.2. 
From  p.  424  of  the  American  Ephemeris  we  find 

Washington  mean  time   T0  =       6h  i8m.4  Y  =  +  .6374 

H=  +  2   36   .9  x   =  +  .5332 

From  table  A,    §252.  8  =  —  15°  33'-4  /=,—  -"73 

log  p  sin  q>  =  9.8112         r0  —  A  =        6h  25™.!  A0  =  2h  43m.6 

log  pcos  <p'  =  9.8810  >^o  =  40°  54' 

Instead  of  computing  at  once  the  values  of  £,  77,  £',  and  rf  with  this  value  of 
.<&„.    let  us   first  determine  the   times  of  immersion  and  emersion  roughly  by 

(429)-(43i). 


sin  A0 
p  cos  <p 

logic 
g  (*'-«') 

log  To 

=  9.8160 
=  9.8810 

cos  7/o 
p  cos  (£>' 
constant  log 

log? 

=  98785 
=  9.8810 

x   -  .5332 
€'  =  .1509 

—  9-4J92 

—  9.6970 

=  9-5824 

=  9.  1787 

=  .1146 

*'-€'  =  -3823 

•r0  = 
The  computation  is  now  as  follows: 


Immersion 

Emersion. 

h 

o  -  2h  43" 

1  6 

£o  =  2h43m.6 

ti 

o  =  i    18 

.1 

r0  =  i    18    .1 

-  30 

+  30 

=  52°  55'  =67°  55' 

We  now 'compute  |,  77,  |',  and  77',  as  follows: 

*  We  might  have  used  Downes's  table  above  referred  to,  where  we  find  re  =  74». 
t  Strictly  TO  should  here  be  reduced  to  sidereal  interval,  but  the  approximation  is  so  rough 
that  it  is  not  important. 


440 


PRACTICAL  ASTRONOMY. 


§255 


sin  d 
cos  d 
sin  ho 

pcos  <f>  sin  d 
cos  ho' 


Immersion. 

9.4284* 

9.9838 

9.9018 

9-3094« 
9.7803 


Check.* 


—    )    =  .0320 

-  ^)'2  =  -0414 
Sum  =  .0734 


log  |  =  9.7828 

p  cos  q>  sin  d  cos  h0'  =  9  0897* 
p  sin  <p'  cos  5  =  9  7950 

log  p  cos  <p'  cos  ho   =  9.6613 
p  cos  <p'  sin  5  sin  //</  =  9.2112* 

log  §'  =  9.0805 
log  TJ'  —  8.6304,, 
sin  M  =  9.8197* 
m  sin  M  = 
m  cos  M  = 


g  =  +0.6064    . 

Nat.  No.  =  —    .1229 
Nat.  No.  =  4-    .6238 

?  =  4-  .7467 

=  *'r  =  4-    .4276 
=  4-    .5433 


-  g)=  -    .1788 
—  77)  =  —     .2034 


tan  J/  =  9.9441 
log  m  =  9.4327 


sin  yv  =  9  9930 
w  sin  A'  =  9.6158 
wcos  AT  =  8.8727* 

tanJV  =    .7431* 
log  w  =  9.6228 

log«'  rr  7.8446 

•in  (M  —  N)  =  9.9328 

log  m  —  9.4327 

i 
log  -   =    .5650 

sin  if;  =  9  9305 

^  =  58°  26'. 2 


=  221°  ig'.2 


I   =  .1204 

rf  —  —  .0427 

*'  =  .5332 

y  =  -  .1173 


x  — 

y  — 


=          .4128 

=  —   .0746 


;V=  100°  14.5 

M  -  N  =  121°    4'.  7 
cos  (J/  —  Af)  =  9.7128* 
logw  =  9.4327 

log-  =  2.1554 

1.3009* 

cos  ^  =  9.7189 
log  k  -  9-4350 

log-,  =  2.1554 
n        

1-3093 


Nat.  No.  =  —  20m.oo 


Nat.  No.  =  ±  20m. 


•39 

Immersion  T\  =  —    o    .39 
Emersion  (inaccurate)  r2  =  -\-  40    .39 
—  A.  =  6h  25m.i 
30m  =  4-  48   .1 
ri  =  -    o  .39 


The  comparison  with  the  true  value  of  £2,  viz.,  .0741,  shows  the  adopted  value  of  h0'  for 


253- 


PREDICTION  OF  AN  OCCULTATION. 


441 


Emersion. 


sin  d  =  9.4284* 
cos  5  =  9.9838 
sin  Ao'  =  9.9669 
cos  <ff  sin  5  =  9.  3094,2 
cos  hj  =  9.5751 

log  £  =  9.8479 


Check.* 


p  cos  <p'  sin 
p  sin 


p  cos  < 
p  cos  <p'  sin 


cos  ^0'  =  8.8845» 
'  cos  d  =  9  7950 


cos  h0  =  9.7561 
sin  h§    =  9.2763;^ 


£  =  +  0.7045 

Nat.  No.  =  —    .0766 

Nat.  No.  =         .6238 

rf  =  +    .7004 


log  r  =  8.8753 
log  rf  =  8.6955« 
sin  M  =  9.8341 
m  sin  M  =  9.4087 
m  cos  M  = 


X   =   XT   =   -\-  .9608 

y  —   Y-\-y'r-=\-  .4260 

(*—€)  =  .2563 

(7  —  */)  =  —  -2744 


tan  M  —  9.9703 
log  m  =  9.5746 


sin  Af  =  9-9953 
n  sin  A^  =:  9.6611 
n  cos  A^  =  8.83o6» 

tan  N  =    .83O5« 
log  n  =  9.6658 
log  n  =  7.8876 
sin  (M  —  A7")  =  9.7946 
Jog  m  —  9.5746 

log-=r     .5650 

sin  iff  =  9.9342 
if)  =  59°  is'o 


M=  136°  57'-6 

£'  =  .0750 

rj'  =  -  .0496 

x'  =  .5332 

/  F  -  -"73 

*'-€'=          .4582 

y  -  V  =  -  -0677 

A7'  =    98°  24'.3 

M-N-    38°33'-3 
cos  (M  —  N)  —  9.8932 
log  m  =  9.5746 

log  —  =  2.1124 
n        

1.5802  Nat.  No.       +  38m.o 

cos  ty  =  9.7087 
log  k  —  9  4350 

log  —  =  2.1124 

1.2561  Nat.  No.         ±  i8m.o 

Emersion  ra  =  —  20   .o 
Immersion  (inaccurate)  TI  =  —  56   .0 
T0  —  A.  =  6h  25m.i 
ro  4"  3om  =  i    48    .1 
r2  =  —20    .01 


7-=  7 


immersion  to  be  nearly  correct.     That  for  emersion,  however,  is  considerably  in  error. 


442  PRACTICAL   ASTRONOMY,  §  254. 

As  a  check  on  the  accuracy  of  these  values  we  now  recompute  x,  y,  £,  and  77, 
when  we  find 

(X  -  ft  +  (y-  ^  =  .07426;  (X  -  ^  +  (y  -  7^   =  .07447. 

i 

We  have  therefore  a  very  close  approximation  to  the  true  time  of  immer- 
sion, the  time  for  emersion  being  a  little  less  accurate.  A  partial  recomputa- 
tion  of  the  latter  gives  a  correction  of  —  om.i6,  making  the  final  value  of 
T  =  7h  53m-°3-  This  latter  computation  is  altogether  unnecessary  for  practi- 
cal purposes. 

For  computing  the  position  angle  P  at  emersion,*  formula  (424),  we  obtain 
a  value  which  will  generally  be  sufficiently  exact  by  using  the  last  values  of 
N  and  T$)  obtained  in  computing  T.  In  this  case  we  have 

N  —    98°  24'  ; 

^  =   59  15  ; 

P  =  N+TJJ+  180°  =  337   39. 


If  the  angle  at  the  vertex  V  is  required,  we  have,  (428)  and  (425), 

tan  C=A±ll;         V=P_C 
7?  +  7/r' 

Using  the  values  just  derived,  viz., 

£  =  .7045,       £'  =  .0750,       77  =  -f  .7004,       77'  =  —  .0496,       ra  =  —  oh.3335, 
we  find  C  =  43°  28'.         Therefore         V  =  294°  u'. 

254.  In  predicting  the  occultations  which  will  be  visible  at  a  given  place 
within  a  given  time,  the  first  operation  will  be  to  go  over  the  list  of  occultations 
of  the  ephemeris  and  select  those  which  may  be  visible.  The  conditions  of 
possible  visibility  are: 

1.  The  limiting  parallels  of  the  last  column  must  include  the  latitude  of  the 
place. 

2.  The  hour-angle  H  —  X,  taken  without  regard  to  sign,  must  be  less  than 
the  semidiurnal  arc  of  the  star;  in  other  words,  the  star  must  be  above  the 
horizon. 

3.  The  sun  must  be  below  the  horizon,  or  at  least  not  much  above  it,  at  the 
local  mean  time  (T  —  A),  unless  the  star  is  bright  enough  to  be  seen  in  the  day- 
time. 

Remark  I.  If  the  place  is  near  one  of  the  limiting  parallels  of  latitude  an  oc- 
cultation  may  or  may  not  occur.  If  it  is  desirable  to  observe  such  stars  as  are 

*This  angle  is  not  required  for  immersion. 


§255-  GRAPHIC  PROCESS   OF  PREDICTION.  443 

occulted  near  the  north  or  south  limbs  of  the  moon,  such  doubtful  ones  may  be 
included  in  our  list,  and  the  occurrence  or  non  occurrence  of  an  occultation  will 
be  shown  in  the  computation  of  the  time  of  immersion  and  emersion.  As  before 
shown,  if  the  occultation  is  not  visible  at  the  place  under  consideration  it  will 

be  indicated  by  sin  ip  becoming  >  I  in  the  formula  sin  i(>  —  —        — . 

Remark  2.  In  most  cases  we  may  see  by  inspection  whether  condition  2  is 
fulfilled.  For  those  stars  near  the  limit  it  may  be  necessary  to  compute  roughly 
the  hour-angle  of  the  star  when  in  the  horizon,  for  which  we  have 

cos  t  =  —  tan  d  tan  <p (122) 

If  then  (H  —  A)  is  numerically  less  than  /  this  condition  is  fulfilled. 

A  small  table  computed  for  the  latitude  of  the  place,  giving  /  with  the  argu- 
ment d,  is  convenient  in  examining  this  condition  and  the  next. 

Remark  3.  For  determining  whether  the  sun  is  above  or  below  the  horizon, 
we  may  compute  roughly  the  times  of  sunrise  and  sunset  by  the  method  given 
above  for  the  star,  or,  since  it  is  not  required  with  great  accuracy,  we  may  take 
it  from  a  common  almanac. 

In  going  over  the  list  of  the  ephemeris,  the  computer  will  write  the  value  of 
A  on  the  lower  edge  of  a  piece  of  paper,  and  pausing  over  each  star  for  which 
condition  i  is  fulfilled,  he  will  see  whether  2  and  3  are  also  fulfilled.  If  either 
fails  the  computer  passes  on.  In  those  cases  where  he  is  unable  to  decide  by 
inspection  whether  either  of  the  two  fail,  the  star  will  be  marked  for  further 
examination  after  the  list  has  been  gone  over. 

Where  many  predictions  are  to  be  made  for  a  given  place  the  work  may  be 
much  reduced  by  computing  tables  for  the  given  latitude  by  means  of  which  the 
computation  of  £,  77,  £',  rf,  and  r  is  facilitated.  The  necessary  directions  for 
forming  and  using  such  tables  are  given  in  the  American  Ephemeris,  to  which 
the  reader  is  referred. 


Graphic  Process. 

255.  If  the  observer  possesses  a  celestial  chart  containing  the  stars  whose 
Qccultation  is  to  be  predicted,  the  necessary  computation  may  be  made  by  a 
very  simple  graphic  process.  The  scale  of  the  chart  must  be  large,  and  the 
method  will  be  principally  useful  in  case  of  clusters  like  the  Pleiades,  where  a 
considerable  number  of  stars  undergo  occultation  within  a  short  time. 

The  right  ascension  and  declination  of  the  moon  are  taken  from  the  ephemeris 
for  intervals  of  half  an  hour  throughout  the  time  covered  by  the  occultations; 
the  correction  for  parallax  must  then  be  applied.  The  resulting  apparent  places 
of  the  moon  are  then  laid  down  on  the  chart,  and  a  curve  being  drawn  through 


444  PRACTICAL   ASTRONOMY.  §  257- 

the  points  we  have  the  apparent  path  of  the  moon's  centre;  this  line  being  then 
properly  subdivided  between  the  half-hour  points  furnishes  a  graphic  time- 
table of  the  moon's  centre.  Each  star  whose  distance  from  this  line  is  less 
than  the  augmented  semidiameter*  of  the  moon  will  suffer  occultation.  From 
such  a  star  as  a  centre,  with  the  moon's  augmented  semidiameter  as  a  radius, 
let  a  circle  be  drawn;  this  circle  cuts  the  path  of  the  moon's  centre  in  two 
points  the  position  of  which  on  the  curve  will  give  the  time  of  immersion  and 
emersion  of  the  star,  and  the  direction  of  the  star  from  the  point  of  intersection 
gives  the  position  angle  on  the  moon's  limb. 


Computation  of  Longitude. 

256.  It  has  now  been  shown  how  we  may  predict  the  time 
of  beginning  and  ending  of  an  occultation,  as  seen  from  a 
point  on  the  earth's  surface  whose  longitude  is  known.  The 
fundamental  equation  which  expresses  the  condition  neces- 
sary for  such  an  occurrence  is 


(415) 


If  now  all  of  the  data  of  the  problem  were  perfectly  known, 
and  if  no  error  entered  into  the  observed  time  of  the  occul- 
tation, this  equation  would  be  completely  satisfied.  Since, 
however,  such  perfection  is  not  attainable,  we  may  employ 
the  observed  time  of  an  occultation  for  determining  the  cor- 
rections to  the  values  of  the  constants  used. 

The  correction  which  it  is  the  immediate  object  of  this 
discussion  to  consider  is  that  of  the  longitude  assumed.  In 
order,  however,  that  this  may  be  obtained  with  all  possible 
precision,  we  must  endeavor  to  obtain  or  eliminate  as  far  as 
possible  the  corrections  to  the  other  quantities  which  enter 
into  the  equation  if  the  values  employed  are  at  all  uncertain. 

257.  Before  making  the  transformation  which  (415)  re- 
quires in  order  to  adapt  it  to  our  purpose,  let  us  examine  the 
quantities  entering  into  each  term  separately,  in  order  to  see 

*  Formula  (392). 


§257-  LONGITUDE  BY  OCCULTATIONS.  445 

what  may  be  regarded  as  definitively  known  and  what 
quantities  may  require  corrections. 

k.  The  moon's  semidiameter  may  be  determined  from  oc- 
cultations  more  accurately  than  in  any  other  way.  A  cor- 
rection Ak  to  the  value  employed  may  therefore  be  intro- 
duced as  one  of  the  unknown  quantities  of  our  equation. 

£,  //.  Referring  to  the  expressions  for  the  value  of  these 
quantities,  equations  (411),  we  see  that  they  depend  upon  a 
and  tf,  the  right  ascension  and  declination  of  the  star  ;  //,  the 
local  sidereal  time ;  p,  the  earth's  radius;  and  cp',  the  geocen- 
tric latitude,  a  and  tf  should  be  so  well  determined  that  they 
may  be  regarded  as  absolute,  that  is,  no  stars  should  be  used 
for  this  purpose  whose  places  are  not  so  well  determined  as 
to  require  no  further  consideration.  >u,  the  local  time,  must 
be  accurately  determined  by  the  transit  instrument  (see 
Chap.  VI).  The  time  determined  by  observation  will  gen- 
erally be  sidereal.  The  ephemeris  of  the  moon  given  in  the 
Nautical  Almanac  is  arranged  for  mean  solar  intervals,  so 
that  when  this  is  employed  it  may  be  necessary  to  convert 
the  sidereal  time  into  mean  solar  time,  or  the  reverse  in  some 
cases.  It  will  be  remembered  that  this  conversion  supposes 
the  longitude  known.  We  shall  therefore  require  an  ap- 
proximate value  of  the  longitude,  which  we  shall  suppose  to 
be  accurate  enough  so  that  no  appreciable  error  will  result 
from  employing  it  for  the  above  reduction.  If  a  case  should 
ever  occur,  which  is  not  likely,  where  this  preliminary  value 
was  so  erroneous  that  appreciable  errors  in  the  subsequent 
computation  resulted  from  its  employment,  then  it  would  be 
necessary  to  repeat  that  part  of  the  computation  which  was 
affected  by  it,  using  the  value  of  the  longitude  obtained  from 
the  first  reduction.  In  this  way  we  should  obtain  a  second 
approximation  to  the  true  value. 

(p.  The  latitude  must  be  well  determined  by  the  zenith 
telescope  or  other  suitable  instrument. 


446  PRACTICAL    ASTRONOMY.  §  258. 

p  depends  upon  the  eccentricity  of  the  earth's  meridian 
passing  through  the  place  of  observation.  A  satisfactory 
determination  of  this  quantity  from  occultations  is  not  pos- 
sible, but  Bessel  introduces  a  term  into  the  equation  depend- 
ing on  the  correction  to  the  assumed  eccentricity,  in  order 
to  show  its  effect  on  the  final  result.  This  term  will  be  re- 
tained for  the  sake  of  completeness,  though  in  the  practical 
application  of  the  formulas  it  will  generally  be  disregarded. 

x  and  y.  Equations  (409).  Besides  quantities  already  con- 
sidered these  contain^,  D,  and  r,  the  right  ascension,  declina- 
tion,  and  distance  of  the  moon.  Corrections  to  the  assumed 
values  of  all  these  quantities  will*  be  introduced  into  the 
equations.  Those  to  the  right  ascension  and  declination  can 
be  well  determined  from  an  occultation  observed  at  any  place 
whose  position  is  known.  In  order,  however,  to  determine 
r,  or  the  moon's  parallax  on  which  r  depends,  observations 
must  be  combined  which  are  made  at  widely  different  points 
on  the  earth's  surface,  whose  difference  of  longitude  has  been 
previously  well  determined.  The  correction  to  the  parallax 
will  be  retained  for  completeness. 

258.  Let  us  now  suppose  a  series  of  occultations  observed 
at  two  points,  the  longitude  of  one  of  which  is  well  deter- 
mined. The  immediate  object  is  to  determine  the  longitude 
of  the  second  point.  If  one  star  only  is  observed  at  the 
second  point,  we  must  assume  all  the  quantities  entering  into 
the  equation  to  be  known  with  one  exception.  If  we  assume 
the  longitude  to  be  the  unknown  quantity,  we  obtain  from 
our  data  a  value  of  that  quantity  which  is  affected  by  all  of 
the  errors  of  the  data.  If  the  star  is  also  observed  at  the 
first  point,  this  observation  may  be  employed  to  correct  the 
tabular  right  ascension  and  declination  of  the  moon,  and  the 
longitude  of  the  second  point  determined  by  the  aid  of  these 
corrected  values.  If  more  stars  are  observed  sufficiently 
near  together  so  that  the  errors  may  be  regarded  as  constant 


§,259-  LONGITUDE  BY  OCCULTATIONS.  447 

during  the  time  elapsed,  then  the  correction  to  the  semi- 
diameter  can  be  included  as  an  unknown  quantity.  As  we 
have  remarked  before,  the  errors  of  the  parallax  cannot  be 
well  separated  from  the  longitude.  If  then  the  number  of 
occultations  observed  is  greater  than  that  of  the  unknown 
quantities  which  can  be  well  determined,  a  solution  of  the  re- 
sulting equations  by  the  method  of  least  squares  will  give 
the  most  probable  values  of  the  quantities,  expressed  in 
terms  of  the  constants,  and  of  those  quantities  which  cannot 
be  separated  from  the  constants. 

259.  We  now  proceed  to  develop  the  equation  in  the  form 
required.  The  method  is  that  of  Bessel.  The  meridian 
from  which  the  longitude  is  reckoned  will  be  called  the  first 
meridian. 

Let          t  =  the  local  time  of  an  observed  occultation — 

mean  or  sidereal ; 

w  =  the  west  longitude  of  the  place  of  observa- 
tion. 
Then  t  -\-  w  =  the  time  at  the  first  meridian. 

Let  r  =  an  arbitrary  time  at  the  first  meridian  suffi- 
ciently near  (/  -|-  w)  so  that  the  change  in 
x  and  y  during  the  interval  (t  -\-  w  —  r) 
may  be  assumed  to  be  proportional  to  the 
time. 

*•„  and  jj/o  are  the  values  of  x  and  y  at  the  time  r. 
Let  Ax,  Ay^  Ak,  be  the  corrections  required  to  reduce  the 
values  of  x,  y,  and  k  employed  to  the  true  .values.     These 
corrections  depend  on  the  various  outstanding  errors  above 
considered. 

The  true  values  of  these  quantities,  corresponding  to  the 
instant  of  observation,  will  then  be 

k  +  Ak', 


PRACTICAL  ASTRONOMY.  §  259. 

x1  and  y'  are  as  before  the  changes  in  x  and  y  in  one  hour, — 
mean  or  sidereal  according  as  one  or  the  other  is  employed. 

Let  Aee  =  the  correction  to  the  assumed  value  of  e* ;  e  be- 
ing the  eccentricity  of  the  meridian. 

Then  £  and  rj  will  require  the  corrections  -y-  Aee  and  —r-Aee. 

dee  dee 

As  these  quantities,  £  and  77,  do  not  depend  upon  the 
longitude,  they  will  be  correctly  given  by  equations  (41 1), 
and  require  no  other  corrections. 

Using  the  corrected  values  of  x,  y,  %,  ?/,  and  k,  equation 
(415)  becomes 


w  - 


-       ee  -  (430 


w  is  supposed  known  with  precision  enough  so  that  the  val- 
ues of  x'  and  y,  which  change  with  the  time,  will  be  known 
with  sufficient  accuracy. 

Let        m  sin  M  =  (x0  —  £);         n  sin  N  =  x'\ 
m  cos  M=  (j0  —  r/);         n  cos  N  =  y'  . 

Equation  (431)  may  then  be  written 

r~ 
—\  ms 


-~4ee    ,    (433) 


§259-  LONGITUDE  BY  OCCULTATIONS.  449 

which  may  be  placed  in  the  form 


*  •intff-  JTH-A*  cosN-  A,  sin  N-  N-*«         (434) 


Let  us  write 


__     d($  sin  ^V+T;  cos  N) 
A  —  Ax  sin  AT+4^  cos  N—  ~          -    .  '  —  ^ 


, 

—  Ar  =  Ax  cosN—4y  sin  ^  --  -  -    —=  —  •  --  -Aee. 

dee 


Then  [k  +  AkJ  =  [n(t+w  -r)+m  cos  (M-N)  +  A]2 

+[**  sin  (Jf_^)_A/]2.  (436) 


Let  msin(M—  N)  =  fcsimp  .....     •     (437) 

Then  neglecting  terms  of  the  second  and  higher  orders  in 
A/  and  Ak,  (436)  may  be  written  as  follows  : 


k  m  •'•  Ak 

t-\-w  —  t  —  -  cos  $  —  -  cos  (M  —  N)-\  --  sec 


A. 

--.      (438) 


TT_  k  m         ,,.  msm(M 

We  have  -  cos  0  --  cos  (M  —  N}  =  - 


n  n  n  sin  ^ 

a  form  which  is  a  little  more  convenient  when  sin  ip  is  not 
very  small. 


450  PRACTICAL   ASTRONOMY.  §  261. 

Equation  (438)  then  gives 


w  = 


sn 


and  the  equation  is  solved  for  w. 

-  As  will  be  seen,  this  value  of  w  is  ambiguous,  ^  being  de- 
termined from  (437)  in  terms  of  the  sine,  with  nothing  to  fix 
the  algebraic  sign  of  cos  i/,\  As  before,  however,  equation 
(423),  the  sign  of  cos  //?  will  be  —  in  case  of  immersion  and  + 
for  emersion.  This  will  always  be  the  case  except  when  the 
occultation  takes  place  very  near  the  north  or  south  limb  of 
the  moon,  when  there  will  sometimes  be  exceptions  to  the 
rule.  Such  occupations,  however,  are  worth  very  little  for 
longitude  purposes,  and  therefore  will  not  require  further 
consideration  here. 

260.  x'  and  j/'  vary  so  slowly  that  the  above  equation  will 
give   a  very  close   approximation  to  the  true  result,  even 
when  (/  +  w  —  r)  is  some  hours  in  duration.     It  will,  how- 
ever, be  best  to  arrange  the  computation  so  that  (/  +  w  —  T) 
is  a  small  quantity,  as  the  labor  is  less  in  dealing  with  small 
quantities  than  with  large  ones,  and  there  is  less  liability  to 
error. 

The  unit  of  time  in  the  small  terms  of  (438)  and  (439)  is  one 
hour.  If  then  w  and  (t  —  r)  are  expressed  in  the  usual  way 
in  hours,  minutes,  and  seconds,  it  will  be  convenient  to  ex- 
press these  small  terms  in  seconds.  If  then  the  time  of  the 
ephemeris  and  of  observation  are  both  sidereal  or  both  mean 
solar,  these  terms  should  be  multiplied  by  3600.  If,  however, 
the  ephemeris  time  is  mean  solar,  and  that  of  observation 
sidereal,  we  must  multiply  by  3609.856. 

261.  Let  us  now  consider  more  fully  the  quantities  A  and  A/. 

These  depend  upon  the  corrections  to  the  moon's  co-ordi- 
nates, viz.,  d&  and  Ay,  and  upon  the  correction  to  the  eccen- 
tricity, Aee.  These  will  be  considered  separately. 


§  26  1  c  LONGITUDE  BY  OCCULTATIONS.  45  1 

The  co-ordinates  x  and  y  are  variable  quantities,  and  the 
corrections  which  they  require  on  account  of  the  inaccuracy 
of  the  data,  viz.,  Ax  and  Ay,  will  also  be  variables.  It  will  be 
more  convenient  for  present  purposes  to  express  these  in 
terms  of  quantities  which  remain  constant  throughout  the 
entire  occultation. 


We  have      x  —  XQ  +  n  sin  N(t  +  w  —  r); 


from  which  we  have 

x  sin  N-\-y  cos  N=     x0  sin  N-\-y0  cos  N-\-n(t-\-w—  T);  )  ', 
—x  cos  N-\-y  sin  N—  —XQ  cos  N-\-y9  sin  N.  f 

The  last  of  these  is  practically  independent  of  the  time, 
and  therefore  may  be  regarded  as  constant  throughout  the 
entire  occultation. 

Let  H  =  —  x0  cos  N  -f-  j/0  sin  N  =  —  x  cos  N  -f-  y  sin  N. 
Then  squaring  and  adding  equations  (441), 

—  r)]2.  (442) 


This  expression  is  a  minimum  when  the  last  term   is  zero. 
Let  the  value  of  (/  -\-  w)  corresponding  to  this  minimum 
be  T.     Then 

XQ  sin  N-\-  j0  cos  N  -\-  n(T  —  ?}  =  o\ 


H  =  —  x  cos  N--      sin  N. 


Therefore  K  —  i/^:2-|-jj/Q  is  the  minimum  distance  of  the 
axis  of  the  cylinder  from  the  centre  of  the  earth,  and  T  is 
the  time  at  the  first  meridian  corresponding  to  this  minimum. 


452  PRACTICAL   ASTRONOMY.  §  262. 

We  now  have  x  sin  N-\-y  cos  N  —  n(t  +  w  —  T)\ ) 

,>    A 7     I       .  .  ~:  ~     A 7"  I**         VI II ) 


Referring  now  to  the  values  of  A  and  A/,  equation  (435),  we 
have  for  the  part  of  these  quantities  depending  on  x  and  y  — 

For  A,        Ax  sin  N  -\-  Ay  cos  N\ 
For  A',  —  Ax  cos  N-\-  Ay  sin  TV. 

Differentiating  equations  (444),  we  have  for  these  quantities 


=  —  nAT+(t  +  w  — 
—  Ax  cos  N  -\-  Ay  sin  N  •=.  An. 

Therefore  that  part  of  the  terms  (Ar  tan  ^  —  A)  due  to  Ax  and 


nA  T-\-  AH  tan  $  —  (/  +  w  —  T)An.     .     .     (445) 

The  corrections  Ax  and  Ay  are  by  this  formula  expressed 
in  terms  of  AT,  AH,  and  An,  which  will  be  constant  for  the 
same  occultation. 

262.  It  remains  to  consider  the  effect  of  an  error  in  the 
eccentricity,  viz.,  Aee,  which  is  considered  here  for  the  sake 
of  completeness,  though  it  might  be  neglected  without  se- 
riously impairing  the  practical  value  of  the  theory. 

From  (134)  and  (140)  we  have 

cos  <p  sin  <p(\  —  ee] 

p  cos  <p'  =      .  --—=--,         P  sin  (pf  =  — =          — ==.  (446) 

V  I  —  ^sm  q>  VI  —  eesin   cp 

dp  cos  q>f        I  dp  sin  <p'         I  , 

-^-  =  -  63  p  cos  <p'\  —j =  -ftp  p  sin  cp'  —  ft. 

dee  2  dee 

p  sin  q>' 
In  which  p 


i  —ee 


§262.  LONGITUDE  BY  OCCULTATIONS.  453 

£         dp  cos  (f 


I     rt  Af|    _  .     _         —  __  -          _____  _  .  _        I        _ 

dee  ~  dp  sin  q>'         dee         '   dp  cos  y>f          dee 


d  v  </?7         - 

dee  ~  dp  sin  cpf         dee         '   ip  cos  q>f          dee 

Referring  now  to  the  values  of  £  and  ?/,  equations  (41 
we  have  . 


7  =—  Sin  dcOS(yW  —  Of); 

' 


d  c,       i  d  77       I 

Therefore  -7-  =  -/fyff£ ;  -7-  =  -/?/?  ?;  —  ft  cos  #.        (447) 

Referring  now  to  the  values  of  A  and  A',  (435),  we  have  for 
the  terms  depending  on  Aee — 


d(i- s'm  N -{- ri  cos  N)  r-      i  „,,,,,    .      ,,.  .  ,_.    ,  ,_-i 

For  A.,—  * bee  = pp(f  sin  Jv-t-  »j  cos  N )  •+•  p  cos  o  cos  N  \&et: 

dee  L      2  J 

For  A',     -          — — —       — bee  —  I |3j3(— ^  cos  j^+ij  sin  ^V)-(-(3  cos  8  sin  A7"  |A*i. 

Let  us  write  ^  =  x^  —  (x^  —  4")  =  ^0  —  ^  sin  J/; 

Substituting  these  values  in  (448)  and  reducing  by  (443), 
find— 


•  •   (448) 


we 


For  A,     -lP3nr-T-mcosM-W          3cosdcosWJee;  )      (      } 

* 


For  A/,  j  —  |/?/?[  K  +  w  sin  (^/ 

We  have  from  (437)  and  (438),  neglecting  the  small  terms  of 
the  latter, 

—  m  cos  (M  —  N)  =  (t  -\-  w  —  -t]n  —  k  cos  ^; 
m  sin  (M  —  N)  =  k  sin  i/>; 


454  PRACTICAL   ASTRONOMY.  §  263. 

which  substitution  will  give  us  for  (449) 


-  \ftft\n(f  +  w  -  T)  -  £cos  £]+  ft  cos  d  cos  N\Aer, 
H      k  sin  yS  cos  d  sin 


\Aer,  \   ,      . 
}  Aee.  f   W  ; 


Therefore  that  part  of  (/V  tan  ip  —  A)  which  depends  upon 
is 


-  T)  -  *  tan  *  -  *  sec  «  -  ^,  .     (45I) 


Therefore  by  (445)  and  (451)  the  last  three  terms  of  equation 
(438)  or  (439)  will  be  as  follows: 

Ak          ,  A'  A  ,  h 

sec  ib  -\ tan  w =  A  T  A tan  ID  AH 

n  '    n  n  '    n 

4-  -  sec  ^  Ak  -  —  (t  4-  w  -  T) 
n  n 

_  ft  cos  8  cos  (vV4-^)~| 

COS  IP 

;  m      —J 

Each  term  is  expressed  in  seconds  of  time,  and  h  is  the  num- 
ber of  seconds  in  one  hour  of  the  kind  of  time  employed  in 
the  ephemeris  of  the  moon.  If  the  times  employed  in  the 
ephemeris  and  in  observation  are  both  sidereal  or  both  mean 
solar,  h  =  3600.  If  the  ephemeris  time  is  mean  solar  and  the 
time  of  observation  sidereal,  h  =  3609.86. 

263.  We  have  now  obtained  an  expression  for  the  small 
terms  of  our  equation,  in  which  the  quantities  depending  on 
the  corrections  to  the  moon's  place  are  expressed  in  terms  of 
quantities  which  are  constant  during  the  time  of  the  occulta- 
tion.  It  will  be  advantageous,  however,  to  express  them 
directly  in  terms  of  the  corrections  to  the  quantities  given  in 
the  ephemeris,  viz.,  to  the  moon's  right  ascension,  declination, 
and  horizontal  parallax. 


§  263 .  LONGITUDE   B  Y  OCCUL  TA  TIONS.  45  5 

Let  A(A  —  d)  —  the  correction  to  the  assumed  difference 

of  right  ascension  of  the  moon  and  star; 

A(D  _  tf)  —  the  correction  to  the  assumed  difference 

of  declination ; 
An  —  the  correction  to  the  assumed  parallax. 

We  have,  equation  (409), 

_  cos  D  sin  (A  —  a)  _  sin  D  cos  8  —  cos  D  sin  8  cos  (A  —  a) 

X  Y 

Writing  for  brevity    x  =  — ,        y  =  — , 

sin  7i  sin  n1 

and  differentiating,  we  have 

AX  An  _AY_          A* 

~  sin  n        tan  Tt'  "  sin  n      Aan  TT* 

These  equations  in  connection  with  (444)  give  the  following: 

-n(t+w-T}-^—  =  -  nAT  +  An(t  +  w  -  T)\ 


sin  Tt  tan  it 

Ait 


sin  n  tan  Tt 


It  will  presently  be  shown  that 
J 


, 
tan  Tt  n  y 


and  therefore 
-  nAT  = 


sin  ._ 

AH  •=  —  _: — L  -  -  _      

sin  TT  tan  n    . 


456  PRACTICAL   ASTRONOMY.  §  264. 

264.  The  value  of  An  will  now  be  more  fully  considered. 

We  have,  equations  (432),  n  sin  N  =  x'\ 

n  cos  N  =  y. 

From  these,  n*  =  x'*  +  y'\ 

Differentiating,        nAn  =  x'  Ax  -\-  y'  Ay  ......     (455) 

x'  andy,  it  will  be  remembered,  are  the  changes  in  x  and  y 
respectively  in  one  hour.  Regarding  them  as  the  differen- 
tial coefficients  of  x  and  y  with  respect  to  the  time,  we  have 


x 

~  x  ; 


dt       </Asin  n    ~'   dt      sin  n 

dy        d(    Y  \  _  dY 

dt 


d(    Y  \  _  dY     i 
~~  afrVsin  7i  1  ~  "  dt  sin  .it  ~~  ?  ' 


-77  and  ~77  depend  upon  the  hourly  change  of  the  moon  in 

right  ascension  and   declination,  which  changes  are  given 
with  accuracy  by  the  ephemeris.     Any  correction  to  the  val- 
ues of  x'  artd  yr  will  therefore  depend  upon  TT. 
We  may  therefore  write 


sin  n  tan  n 

b  An 


Ay'  =  A-. =  -  y' 

<r  cir»      TT  <r 


sm  n  '  tan  n 

Substituting  in  equation  (455),  it  becomes 


. 
tan  n 

Therefore  —  —  —  7  -  ,  the  value  assumed  above. 

n  tan  n 


§265.  LONGITUDE  BY  OCCULTATIONS.  457 

265.  Returning  now  to  equations  (454),  we  see  that 

An  AX  AY 

and 


tan  TCJ        sm  TT'  sin  7t 

may  be  regarded  as  constant  throughout  the  duration  of  the 
occultation,  since  they  are  expressed  in  terms  of  AT  and 
JK,  which  are  constant,  and  An  and  A7,  which  are  practically  so. 

The  values  of  -.  -  and  -.  --  will  then  result  from  the  differ- 
sin  n         sin  n 

entiation  of  equations  (453),  viz.: 

X  =  cos  D  sin  (A  —  «); 
Y  —  sm  D  cos  6  —  cos  Z>  sin  d  cos  (/2  —  or); 
AX  =  cos  /}  cos  (A  —  a)A(A  —  a)—  sin  D  sin  (^4  —  «)  J/7; 
A  Y  =  [cos  Z?  cos  tf  +  sin  £>  sin  6  cos  (^  —  a)]AD 
+  cos  Z?  sin  tf  sin  (^  —  a)  A  (A  —  a) 
—  [sin  Z>sin  d  +  cos  Z>  cos  d  cos  (^4  —  a)]^tf. 

At  the  time  of  conjunction  of  the  sun  and  moon  A  be- 
comes equal  to  a.  Therefore 

AX       cos  D  .  AY       cos  (D—d) 

-T  --  =  —.  ---  A(A  —  a)-         -  --  —  -  —+  A(D—  ^.(456) 

sm  7t       sin  n  sin  n  sin  n 

Therefore  taking  D  and  n  for  the  instant  of  conjunction  of 
the  moon  and  star  in  right  ascension,  and  regarding  A(A  —  ot) 
and  A(D  —  tf)  as  the  corrections  to  the  assumed  differences 
of  right  ascension  and  declination  at  this  instant,  also  writing 
unity  for  cos  (D  —  tf),  n  for  sin  n  and  tan  TT,  we  have,  from 

(454), 

cos  DA  (A  -a)   .  A(D-d} 


—AT= 

rnt 


A(D-d~)    ,  A* 

-  -  LsmN—Ji  —  ; 
n  ft 


An         An 
n  n  * 


-  (457) 


458  PRACTICAL  ASTRONOMY.  §  266. 

Substituting  these  values  in  (452),  and  writing  for  brevity 

"  =  ^  ........    (453) 

we  have 

—  sec  VH  --  tan  ^  --  =  —  v[sin  A^cos  DA(A  —  a)-j-  cos  NA(D  —  8)] 

-\-v[-  cos  NCOS  DA(A  -  a)  +  sin  NA(D  -  d)] 
-f-  v  sec  if>  7T/U  -f-  y[«(/  -j-  w  —  71)  —  «  tan  ^]//?r 

(459) 


This  equation  gives  the  expression  for  the  last  three  terras 
of  (438)  or  (439),  in  which  An  and  Aee  are  completely  sepa- 
rated from  the  other  corrections. 

266.  Let  us  now  write 


— I 

-  cos  ib cos  (M  —  N)    —  (/  — 

n  n 


y  =  sin  TV  cos  DA(A  —  a)  -\-  cos  NA(D  —  tf); 
3  =  —cos  NCOS  DA(A  —  a)  +  sin  NA(D  —  (J); 
E  —  n(t  +  w  —  T)  —  H  tan  ^; 


rT 

=       - 

|  _  2 


/T       /^  COS  (5  COS 

w—  7^)  —  «tan  ^—  ^sec^J  ----  -  -  l—    \7t. 

2  COS  ^ 


^(460) 


Then  equation  (438)  becomes 

iv  =  £1  —  vy  -f-  v  tan  $5  -{-  v  sec  tyTtAk  -\-  vEAit  -f-  vFAee.  (461) 

This  equation  is  now  in  a  form  which  is  well  adapted  to 
the  purpose  in  view. 

w,  y,  £,  TtAkj  An,  and  /W  may  in  certain  cases  be  treated 
as  unknown  quantities,  but  they  can  never  all  be  determined 
at  the  same  time  from  the  same  series  of  equations. 

vy  is  a  constant,  and  its  value  is  independent  of  the  longi- 
tude of  the  place  of  observation.  In  order  to  make  its  de- 


§267.  LONGITUDE  BY  OCCULTATIONS.  459 

termination  possible,  therefore,  the  occultation  should  be  ob- 
served at  one  place  at  least  whose  longitude  is  known.  In 
case  such  an  observation  is  not  available,  y  may  be  deter- 
mined from  meridian  observations  of  the  moon,  if  such  are 
available,  made  on  the  same  night  or  sufficiently  near  the 
same  time  that  AA  and  AD  may  be  well  determined  from 
them.  Of  course  if  the  ephemeris  of  the  moon  were  perfect 
this  would  be  unnecessary,  as  then  A A  and  AD  would  be 
zero. 

267.  In  case  simply  the  immersion  or  emersion  of  a  star  has 
been  observed  at  two  places,  the  longitude  of  one  of  which 
is  well  determined,  the  power  of  the  data  will  be  exhausted 
with  the  determination  of  w  and  y.  If  both  the  immersions 
and  emersions  have  been  observed,  we  may  also  determine 
nAk  and  5  as  unknown  quantities,  but  in  no  case  can  An  be 
determined  from  occultations  unless  w  has  been  previously 
well  determined.  Still  less  can  a  satisfactory  determination 
oidee  be  obtained  in  this  manner.  The  two  last  terms  may, 
however,  be  retained  in  the  solution  of  the  equations  in 
order  to  show  the  effect  on  the  resulting  longitude  of  an 
error  in  n  or  in  ee.  At  the  same  time  it  will  make  it  possi- 
ble to  apply  the  necessary  correction  to  the  longitude,  if  from 
any  source  values  of  these  quantities  become  known  more 
accurate  than  those  assumed  in  the  computation. 

For  the  determination  of  Ak  from  single  occultations  both 
immersion  and  emersion  must  be  observed,  but  contacts  at 
the  bright  limb  can  be  observed  much  less  satisfactorily  than 
at  the  dark  limb. 

The  best  results  are  obtained  from  the  occultations  of 
groups  of  stars  like  the  Pleiades,  in  which  the  relative  posi- 
tions of  the  stars  are  well  determined.  The  passage  of  the 
moon  through  such  a  group  furnishes  a  number  of  equations 
ot  condition  of  the  form  (461),  equal  to  that  of  the  observed 
disappearances  or  reappearances  of  the  stars  occulted.  As 


4^0  PRACTICAL   ASTRONOMY.  §  268. 

before  remarked,  observations  at  the  dark  limb  can  be  made 
with  much  greater  accuracy  than  at  the  bright  limb  (except 
perhaps  in  case  of  a  few  of  the  brighter  stars).  If  it  is 
thought  desirable,  therefore,  only  observations  made  at  the 
dark  limb  need  be  used  in  the  equations,  especially  so  if  stars 
are  observed  both  north  and  south  of  the  moon's  equator. 

On  account  of  the  advantages  offered  by  the  Pleiades  for 
this  purpose,  Prof.  Peirce  developed  the  equations  in  a  form 
especially  adapted  to  this  group,  for  use  in  the  longitude 
work  of  the  U.  S.  Coast  Survey.  The  reader  who  is  suffi- 
ciently interested  in  the  subject  may  refer  to  the  reports  of 
the  U.  S.  Coast  Survey,  1855-56-57-61,  in  the  latter  of  which 
is  given  a  numerical  example  of  the  application  of  the  method. 


Correction  for  Refraction  and  for  Elevation  above  Mean  Sea 

Level. 

268.  The  fundamental  equation  which  has  been  used  as  the 
basis  of  our  analysis  expresses  the  condition  that  the  point 
from  which  the  immersion  or  emersion  is  observed  is  situ- 
ated in  the  surface  of  a  right  cylinder  enveloping  the  moon 
and  star.  At  the  same  time  it  has  been  supposed  to  be  in 
the  spheroidal  surface  of  the  earth. 

The  refraction  which  the  ray  suffers  in  passing  through 
the  atmosphere  causes  the  elements  of  this  cylinder  to  be 
curved  lines  instead  of  right  lines ;  or,  more  correctly,  the 
surface  is  not  that  of  a  cylinder.  Further,  it  follows  from 
the  irregularities  of  the  earth's  surface  that  the  point  from 
which  the  observation  is  made  will  not  in  general  be  in  the 
surface  of  the  mean  ellipsoid.  Neither  of  our  surfaces  there- 
fore conforms  exactly  to  the  mathematical  form  assumed. 
The  effect  upon  the  observed  time  of  an  occultation  will 


§  268.  LONGITUDE  BY  OCCULTATIONS.  461 

always  be  small,  but  in  extreme  cases  must  be  taKen  into  ac- 
count in  an  accurate  investigation. 

If  we  consider  a  ray  of  light  as  it  comes  to  the  eye  at  the 
instant  when  the  star  is  apparently  in  contact  with  the  moon's 
limb,  this  ray  will  form  a  curved  line,  the  asymptote  of  which 
will  cut  the  vertical  line  of  the  observer  at  a  point  where  the 
contact  would  be  seen  at  the  same  instant  as  that  observed 
if  no  refraction  existed.  The  effect  of  refraction  will  then 
be  taken  into  account  if  we  substitute  this  point  for  the  point 
occupied  by  the  observer. 

Let          ti  =  the  altitude  of  this  fictitious  point  above  the 

observer's  position ; 
h  •=.  the  altitude  of  the  observer's  position  above 

the  mean  sea  level. 

Then  h  +  h'  —  the  altitude  of  the  fictitious  point  above  the 
mean  sea  level. 

Let  us  then  suppose  the  observation  to  be  made  from  a 
point  at  this  elevation  above  the  surface  of  the  mean  ellipsoid. 

The  necessary  transformation  will  be  accomplished  by 
changing  p  cos  cp'  and  p  sin  cp'  into  p  cos  <p'  -f-  (h  +  h')  cos  cp 
and  p  sin  cpf  -\-  (h  -|-  h'}  sin  cp ;  or,  by  formulae  446, 


p  cos  cp'  [i  +  (h  +  h')   1/1  —  ee  sin2  cp] 
and  prin 


h  and  h'  will  always  be  very  small  fractions  when  ex- 
pressed in  parts  of  the  earth's  radius  ;  therefore  no  apprecia- 
ble error  will  result  from  neglecting  the  products  of  these 


PRACTICAL   ASTRONOMY.  §  268. 

quantities  by  ee.  Also  (\-\-h-\-  k'}  will  be  practically  equal 
to  (i  +  ti)  (i  +  h'\  the  small  term  hh'  being  of  no  account. 

The  necessary  correction  for  elevation  above  the  mean 
sea  level  will  therefore  be  obtained  by  adding  to  log  p 
log  (i  +^),  and  the  correction  for  refraction  by  adding 
log  (i  +  h'\ 

Expanding  log  (i  -f-  ft),  we  have 


log  (i+A)  =  M(h-  J+etc.) 


M  =  .43429448  is  the  modulus  of  the  common  system  of 
logarithms. 

h  is  here  expressed  in  terms  of  the  earth's  radius.     If  it  is 

given  in  feet  we  shall  have,  instead  of  the  above,  -  . 

20923597 

Therefore,  neglecting  squares  and  higher  powers  of  //, 


+  ^)  —  ^(.ooo  ooo  02076).    .    .     .     (462) 


If,  for  instance,  the  elevation  is  1000  feet,  the  correction 
to  be  applied  to  log  £  and  log  rj  will  be  .0000208. 

The  factor  (i  +  ^')  will  now  be  considered. 

In  the  general  theory  of  refraction  the  atmosphere  is  re- 
garded as  composed  of  concentric  strata  the  thickness  of 
which  is  uniform  and  may  be  regarded  as  infinitesimal.  If 
the  distance  of  any  point  in  a  ray  of  light  from  the  earth's 
centre  be  r,  i  the  angle  between  the  tangent  and  normal  at 
the  point  to  which  r  is  drawn,  then  it  is  shown  by  the  theory 
of  refraction  that  pr  sin  i  is  a  constant,  ^  being  the  index  of 
refraction  for  the  infinitesimal  stratum  at  the  point  under 
consideration. 


268. 


LONGITUDE  BY  OCCULTATIONS. 


463 


For  the  point  where  the  ray  enters  the  eye  let  r0,  yw0,  and 
z'  be  the  special  values  of  r,  /*,  and  i.  Then  zf  will  be  the 
apparent  zenith  distance  of  the  star,  and  from  the  foregoing 


0  sin  z'  = 


sn   . 


(463) 


If  the  first  point  is  taken  so  far  away  as  to  be  beyond  the 
limit  of  the  earth's  atmosphere,  then  the  refraction  at  this 
point  is  zero  and  //  becomes  unity.  z 

The  above  equation  then  becomes 

//Or0  sin  2  =  r  sin  i.  .     (464) 
In  the  figure, 

OP  =  r0;  PQ  =  h'\ 

Or  =  r\  OrQ  =  i. 

ZQr  =  z  is  the  true  zenith  dis- 
tance of  the  star  observed. 
Then  from  the  triangle  rQO 


(rc  -j-  h')  sin  z  =  r  sin  i ; 
and  from  equation  (464) 

(r.  -\-  hf)  sin  2  =  /v0r0  sin  z' , 


from  which 


h'  sin  z' 

=  un  - 


sm  z 


r0  will  not  differ  appreciably  for  this  purpose  from  the 


464 


PR  A  C  TIC  A  LAS  TR  ONOM  Y. 


§268. 


equatorial  radius  of  the  earth ;  so  that  if  we  regard  h'  as  ex- 
pressed in  terms  of  this  quantity  we  have 


log  (i 


i        Sin  ^r 

log  -     -  +  log  /IQ.  . 


sin  z 


(465) 


The  mean  value  of  //0  is  i.ooo  2800. 

A  table  is  readily  arranged  for  log  (i  +  h'\  with  the  argu- 
ment #,  the  zenith  distance  of  the  star.  By  referring  to  the 
value  of  8, — equations  (41 1) — we  see  that  £  is  very  nearly  equal 
to  cos  2.  For  this  purpose  we  may  consider  it  the  same. 

The  following  is  Bessel's  table  for  log  (i  +  h').  In  addi- 
tion to  the  argument  z  we  have  given  cos  z,  for  which  we 
may  use  log  <?  without  appreciable  error. 


TABLE  B. 


* 

log  cos  z 

log(i+A') 

z 

log  cos  z 

log  (i  4-  h') 

o° 

.0000 

0.0000000 

82°  o' 

9.1436 

o  .  0000069 

10 

9-9934 

o  .  ooooooo 

83   o 

9.0859 

O.OOOOO86 

20 

9.9730 

o  .  ooooooo 

84  o 

9.0192 

O.OOOOIII 

30 

9  9375 

0.000000  I 

85  o 

8  .  9403 

0.0000147 

40 

9-8843 

O.OOOOOOI 

85  30 

8.8946 

0.0000169 

50 

9.8081 

O  .  OOOOOO2 

86  o 

8  .  8436 

0.0000198 

60 

9  .  6990 

o  .  0000005 

86  30 

8.7857 

0.0000234 

62 

9.6716 

o  .  0000006 

87  o 

8.7188 

0.0000280 

64 

9.6418 

o  .  0000007 

87  30 

8.6397 

o  0000337 

66 

9  •  6093 

0.0000008 

88  o 

8  5428 

0.0000412 

68 

9-5736 

o  .  0000009 

88  30 

8.4179 

o  .  00005  i  i 

70 

9-5341 

O.OOOOOI2 

88  50 

8.3088 

0.0000594 

72 

9  .  4900 

0.0000015 

89  oo 

8.2419 

o  .  0000643 

74 

9.4403 

0.0000019 

89  10 

8.1627 

o  .  0000695 

76 

9-3837 

0.0000025 

89  20 

8.0658 

0.0000753 

78 

9-3179 

0.0000033 

89  30 

7.9408 

o  .  00008  i  7 

80 

9-2397 

o  .  0000046 

89  40 

7.7648 

0.0000888 

81 

9.1943 

0.0000056 

89  50 

7.4637 

0.0000967 

82 

9ij."}6 

o  .  0000069 

QO  OO 

o  .  0001054 

•  i'tj^' 

TT* 

§  268.  LONGITUDE  BY  OCCULTATIONS.  465 

Example.  The  following  occultations  of  stars  of  the  Pleiades  group  were 
observed  at  Washington  and  Greenwich  on  September  26,  1839: 

AT  GREENWICH.  AT  WASHINGTON. 

Star.                          Sidereal  Time.  Sidereal  Time. 

^-Ceheno 5h  23""  538-85  22h  5im  ig'.gg 

^Taygeta 5    56    50.63  23      i      0.68 

c  Maja 5    58    17-43  23    17    46.52 

These  are  all  emersions  observed  at  the  dark  limb  of  the  moon. 
The  observations  at  Washington  were  made  at  Gilliss's  observatory  on  Capitol 
Hill,  the  position  of  which  is  assumed  to  be  Latitude  (p  =  38°  53'  32". 8 

West  longitude  5h    8m  i8.75 

The  latitude  of  Greenwich  q>  —  51°  28'  38''.4 

We  now  take  from  Bessel's  catalogue  of  the  Pleiades  the  right  ascensions 
and  declinations  of  the  stars  for  1839.0  and  reduce  them  to  apparent  place  for 
1839,  September  26,  Greenwich  3h  and  6h  sidereal  time,  viz.: 

a  3h  a  6h  5  3"  8  6h 

g  Celaeno. . .   53°  49'  34"-68       53°  49'  34". 72          23°  46'  56". 47       23°  46'  s6".48 
^Taygeta...   53    55   27  .47       53    55   27  .51  23    57  40  .96       23    57  40  .97 

c  Maja 54     4  47  .27       54     4  47  .31          23    51   50  .01       23    51   50  .02 

The  right  ascension,  declination,  and  horizontal  parallax  of  the  moon  for  four 
consecutive  hours — viz.,  3h,  4'',  5h,  and  6h  Greenwich  sidereal  time — are  as 

follows: 


3h.. 

*  Moon's  A 

D 

2d°  8'  <ft"  O7 

7T 

60'  10"  19 

4h 

53  18  58  26 

24  1  8  44  85 

60  8  88 

24  28  24  .41 

60   7^7 

6h.. 

.  54  ^6  8  .oq 

24  ^7  «  .7-3 

60  6  .2* 

We  now  compute  x  and  y  for  these  dates  for  each  of  the  stars  from  formulae 
(410),  viz., 

_  cos  D  sin  (A -a)  p          _  sin  (Z>-8)  cos2  j(A-a)  -f  sin  (D  -f  8}  sin2  \(A  -  a\ 
sin  n  sin  it 

*  These  values  are  given  by  Peirce,  Coast  Survey  Report  1861,  pp.  204,  205.  They  were  com- 
puted directly  from  Hansen's  tables.  When  the  Nautical  Almanac  is  used  the  intervals  will  be 
mean  solar  hours. 


466  PRACTICAL   ASTRONOMY. 

The  computation  is  given  in  full  for  g  Celaeno. 


§268. 


log  it 
S 
sin  it 

3h 

4h 

5h 

6* 

3.5575301 

4.6855527 

8.2430828 

3.5573724 
4.6855527 
8.2429251 

3.5572I48 
4.6855527 
8.2427675 

3.5570558 
4.6855527 
8.2426085 

cosec  it 

1.7569172 

1.7570749 

L7572325 

i  .75739*5 

A 
a 
A  -  a 

sin  (A  -  a)  j 

cosZ> 
cosec  it 
log* 

52°  40'  29".  52 
53  49  34  .68 
-i   9  5  -16 
4-6855457 

3.6i754i3n 
9.9602268 
1.7569172 
.  02023  ion 

53°  18'  s8".26 
53  49  34  .69 
-o  30  36  .43 
4.6855692 
3.2639744,^ 
9.9596679 

1.7570749 
9.6662864n 

53°  57'  3i".09 
53  49  34  .70 
+  o   7  56  .39 
4-6855745 
2.6779626 
9.9591146 

1.7572325 
9.0798842 

54°  36'  8".  03 
53  49  34  -72 
+  o  46  33  .31 
4.6855616 
3.4461191 
9.9585670 

I.75739I5 
9.8476392 

X 

—  1.047686 

-0.463753 

+0.120194 

+0.704108 

D 

d 
D-d 
D  +  d 
HA  -  a) 

24°  8'  55".07 
23  46  56  .47 

0  21  58  .60 

47  55  5i  -54 
-  34  32  .58 

24°  18'  44".  85 
23  46  56  .47 
o  31  48  .38 
48   5  4i  -32 

—  15  l8  .22 

24°  28'  24".  41 
23  46  56  .48 
o  41  27  .93 

48  15  20  .89 

+  3  58  .20 

24°  37'  53".73 
23  46  56  .48 
o  50  57  -25 

48  24  50  .21 

-f  23  16  .66 

sin  \(A  ~  a)  j 

sin2  \(A  -  a) 
sin  (D  +  d) 
Sum  i 

4.6855676 

3-3165113 
6.0041578 
9.8706018 
5.8747596 

4.6855735 
2.9629467 

5  .  2970404 

9.8717195 
5.1687599 

4.6855748 
2.3769418 
4.1250332 
9.8728115 
3.9978447 

4.6855716 
3.1450906 
5.6613244 
9.8738782 
5.5352026 

cos9  \(A  —  a) 
sin  (D  -  5)  | 
Sum  2 

9.9999562 
4-6855719 
3.1201131 
7.8056412 

9.9999914 
4.6855687 

3.2806649 
7.9662250 

9.9999994 
4.6855644 
3.3958381 
8.0814019 

9.9999798 
4.6855590 
3.4853310 
8.1708698 

52  -  Si 

Zech* 
cosec  TT 
log^/ 

1.9308816 
.0050625 
1.7569172 
9.5676209 

2.7974651 

.0006918 

1.7570749 
9.7239917 

4.0835572 
.0000358 

1.7572325 
9.8386702 

2.6356672 
.0010038 

I.75739I5 
9.9292651 

y  = 

+  -369506 

.529653 

.689716 

.  849699 

*  This  is  the  quantity  taken  from  Zech's  addition  and  subtraction  logarithmic  table. 


268. 


LONGITUDE  BY  OCCULTATIONS. 


467 


We  thus  have  values  of  x  and  y  computed  for  four  consecutive  hours,  from 
which  we  can  now  compute  the  values  of  x'  and  y'  to  the  third  order  of  dif- 
ferences inclusive  by  means  of  formulae  (101),  (101)1,  and  (ioi)2,  viz. : 


X 

3h  —  1.047686 
4h  -  .463753 
5h-f-  .120194 
6h-f-  .704108 

X1 

.583910 
.583948 
.583938 
.583882 

9 

.369506 

•529653 
.689716 
.849699 

y' 
.160189 
.160105 
.160023 
.159941 

For  the  other  stars  observed  we  find — 


Taygeta. 


X 

3h  —  1.136840 

4h  -  -552839 
5b  +  .031182 
6h+  .615185 

x' 

+.583978 
.584017 
.584018 

.583981 

y 
+.191768 

•351424 
.511015 
.670546 

y' 

.159690 
.159623 
.159560 

•159503 

Maja, 


3h  —  1.278300 

+.584071 

.290289 

.159105 

4h  -  .694197 

.584128 

•449353 

.159024 

5h  —  .110057 

.584145 

.608340 

•158951 

6h+  .474080 

.584122 

•767257 

.158884 

Computation  of  £j,  rf,  and  C. 

(C  is  only  required  for  determining  the  correction  due  to  refraction.) 
Formulae  (412)  are  as  follows: 


p  sin  q>  =  b  sin  /?; 
p  cos  <p'  cos  (JJL  —  a)  —  b  cos  B; 


=  p  cos  q>  sin  (//  —  or); 

=  b  sin  (B  -  5); 
=  b  cos  (B  -  8}. 


With  the  known  values  of  (p  for  Greenwich  and  Washington,  we  obtain  p 
and  cf)  by  the  use  of  formulae  (V),  Art.  77. 


.  PRACTICAL   ASTRONOMY. 

The  computation  is  then  as  follows: 


§268. 


Greenwich. 

Washington. 

9 

51°  17'  24".  8 

38°  42'  i8''.3 

sin  cp 

9.8922748 

9.7960967 

log  p 

9-999II35 

9.9994302 

cos  cp 

9.7961411 

9  8923033 

p  sin  cp 

9.8913883 

9.7955269 

p  cos  cp 

9.7952546 

•9  8917335 

A 

5h  23™  53s.  85 

22h  51'"  19s.  99 

M 

80°  58'  27".8 

342°  49'  59".  9 

a 

53   49    34  -7 

53    49  34  -7 

V  —  a 

27     8    53  .1 

289      o  25  .2 

cos(#  —  a] 

9.9493072 

9.5127960 

sin  (/*  —  or) 

9.6592427 

9.9756518,* 

log? 

9.4544973 

9-8673853« 

i 

+.284772 

—  .736861 

£cos  j? 

9.7445618 

9.4045295 

sin  .Z? 

9  9107179 

9.9668001 

£  sin  j? 

9.8913883 

9.7955269 

tan  B 
B 

.1468265 
54°  30'  2i".2i 

•3909974 
67°  52'  51".  33 

8 

23  46  56  .47 

23    46  56  .47 

B-  d 

30    43  24  .74 

44      5    54-86 

sin  (B  —  8) 

9.7083326 

9.8425436 

log£ 

9.9806704 

9.8287268 

cos  (£-8) 

9.9343179 

9.8562113 

log  77 

9.6890030 

9.6712704 

77 

.488656 

.469105 

logC 

9  9149883 

9.6849381 

z 

34°  4i' 

61°    3 

z  has  been  computed  for  the  purpose  of  taking  into  account  the  correction  for 
refraction.  With  this  value  we  find  from  table  B,  Art.  268,  log  (i  -f-  A')  = 
.0000001  and  .0000005  respectively,  which  values  are  to  be  added  to  log  | 
and  log  77.  As  they  are  so  small  as  to  be  practically  inappreciable,  they  have 
been  neglected. 

Also,  we  have  for  the  above  times  of  observation — 

TAYGKTA.  MAJA. 

Greenwich.          Washington.  Greenwich.         Washington. 

€+.360523          —.725974  4-302353          —.704226 

7/+. 504728          +.455553  +.506584          +.436040 


§268. 


LONGITUDE  BY  OCCULTATIONS. 


469 


With  the  assumed  value  of  the  longitude  of  the  observatory  at  Washington, 
viz.,  5h  8m  Is. 75,  we  reduce  the  Washington  times  to  Greenwich  time,  and  as- 
suming the  values  of  T  sufficiently  near  these  times  that  x  and/  may  be  assumed 
to  vary  uniformly  during  the  interval,  we  compute  M,  m,  N,  n,  and  if)  by  the 
formulae 


m  sin  M  =  XQ  — 
m  cos  M  =  jj/o  — 


n  sin  JV  =  x'  \ 
n  cos  N  =  y' ; 


=  —  sin  (M  —  N\ 


The  computation  for  Celaeno  is  then  as  follows: 


Wash,  time 
Gh.  time 
Assumed  T 

Greenwich. 

Washington. 

5h  23m  538.8s 
5".4 

22h  5im  i98-99 
3    59     21  .74 
4h.o 

Xo 

1 
*o-| 

•353765 
.284772 
.068993 

-.463753 
-.736861 
+•273108 

Jo 

V 

yo  —  ?/ 

•753720 
.488656 
.265064 

•529653 
.469105 
.060548 

log  m  sin  M 
sin  Af 
log  m  cos  M 

8.8388050 
9.4012192 
9.4233508 

9.4363344 
9.9895810 
8.7820998 

tan  J/ 
/]/ 
log  m 

9.4154542 

14°  35'  22".  8 
9-4375858 

.6542346 
77°  29'  59".o 
9.4467534 

X1 

y 

.583916 
.159990 

.583948 
.160105 

log  n  sin  yV 
sin  JV 
log  n  cos  ^ 

9.7663504 
9.9842810 
9.2040928 

9.7663742 
9.9842609 
9.  2044049 

tan  JV 
N 
log  w 
M  -  N 
sin  (^/  -  yV) 
log  m 
ac  log  £ 
sin  if} 

.5622576 
74°  40'  38".  3 
9  7820694 
299°  54'  44".  5 
9  9379T35« 
9  4375858 
.5650000 
9.9404993^ 

.5619693 
74°  40'    3".4 
9.7821133 
2°  49'  55".6 
8.6938108 
9.4467534 
.5650000 
8.7055642 

4/o 


PR  A  C  TIC  A  L   AS  TJRONOM  Y. 


268. 


Since  the  emersions  were  the  phases  observed,  cos  ip  is  plus;  therefore 

Greenwich. 
ip=  299°  i8'43".7 

We  now  compute  fl  from  the  formula 


Washington. 

2°  54'  35"-5. 


£1  =  h\  k-  cos  V  -  ^  cos  (A/  -  710 


where 


=  3600; 


-  (/  - 
log  h  -  3.5563025. 


COS  ^ 

log± 

Nat.  No! 

Greenwich. 

Washington. 

9.6898123 
9.4350000 

.2179306 

2.8990454 
792".  58 

9-9994397 
9.4350000 

.2178867 

3.2086289 
i6i6s.7o 

—  cos  ^ 
n 

cos(M  -  N} 
log  m 

log  - 

/7 

Nat.  No! 

I3m  12".  58 

26m  56".  70 

9.6978174 

9-4375858 

.2179306 

2.9096363 
8i2M5 

9.9994692 
9.4467534 

.2178867 

3.2204118 
i66iM6 

t—  T 

13™  32s.  15 
-    6.15 

27m  4is.i6 

—  5h  8m  40s.oi 

a 

-  1  3".  42 

45"  7m  55s-55 

In  a  similar  manner  we  find  for  the  other  stars — 

ForTaygeta,  fl        —    9".  30        +5h  7m  558-67; 

For  Maja,  fl         -    9*.  79         +5h  7m  538-o8. 

We  next  compute  T,  H,  and  v  by  formulae  (443)  and  (458),  viz.: 
T  =  r (x0  sin  N  -f  jo  cos  N); 

H  =  —  x»  cos  N  -\-  }'0  sin  N\ 

h 

v  =  — . 
mi 

*  It  is  not  necessary  for  this  purpose  to  know  the  value  of  k  with  extreme  accuracy,  since 
the  correction  A£  to  the  assumed  value  appears  as  one  of  the  terms  of  our  equation. 


§268.  LONGITUDE  BY  OCCULTATIONS. 

For  Celaeno  we  have 


*  5-4 
sin  ^9.98428 

Iogx0  9.54871 

cos  tf  9  42202 
log  jo  9.87721 


log 


log  *<>  cos  N  8.97073 

Zech    .83098 
log  jo  sin  N  9.86149 

log  n  9.80171 
*    -6334 


471 


log-  6.44270 

log^-    .21793 
log^  3.55630 

log  v    .21693 
v  1.6479 


Iog(jf0  sin 


sin  ^9.53299 
Zech  .43345 
log  jo  cos  ^9.29923 

A^-f-jo  cos  A7")  9  73268 
log  l-  .21793 

9  95061 

Nat.  No.  .8925 
^4.5075 

We  now  compute  the  coefficients  for  the  final  equations  of  the  form  (461),  viz.: 
v  tan  i(>,         vE  =  v\n(t  -\--w-  T)  —  H  tan  ifi\t          and         v  sec  ^. 


Greenwich. 

Washington. 

/+  w 

5  3983 

3.9894 

t-\-  w  -  T 

.8908 

—  .5181 

log  (t+w-  T) 

9.94978 

9-7i44i» 

logn 

9.78207 

9.78211 

Sum 

9-73I85 

g.49652« 

log  u 

9.80171 

9.80171 

tan  $ 

.25o69» 

8.70612 

Sum 

.05240^ 

8.50783 

Zech 

.16969 

.04241 

log^ 

.22209 

9-53893» 

log  v 

.21693 

.21693 

log  v  E 

.43902 

9-75586» 

vE 

2.7480 

-.5700 

sec  ^ 

.31019 

.00056 

log  v  sec  ip 

.52712 

.21749 

log  v  tan  ^ 

.46762^ 

8  92305 

v  sec  #> 

3.3661 

1.6500 

v  tan  if)\  —  2.9351 

.0838 

47  2  PRACTICAL  ASTRONOMY.  §  268. 

Computing  the  coefficients  for  the  other  two  stars  in  the  same  way,  we  ob- 
tain the  following  six  equations: 


Celaeno:  G.  w  =  —  oh  om  i38.42  —  1.6487  —  2.9351?  -f  ^.^66rr^  +  2.748A7T;  [i] 

W.  w'  =  5  7  55  -55  -  1-648?  +  .0841?+  i.6soirAA  -  .57oA»r.  [4] 

Taygeta:  G.  w  =  -  o  o  9  .30  -  1.6487  -  .5981?  +  i.7537rA£  -f  i-so/Aw;  [2] 

W.  «/  =  57  55  -67  -  1-6487  +  1.048*  +  i.953«**  -  i.o84Air.  [5] 

Maja:  G.  w  =  —  o  o  9  .79  —  1.6487  —  2.328*  -\-  2.8527rA/t  f-  2.492A?r;  [3] 

W.  TV'  =  57  53  .08  —  1.6487  —  .062*+  i.6so7rA/&  —  .442A7T.  [6] 


(A) 


If  we  assume  y,  3,  Ait,  and  itAk  to  be  the  same  in  all  of  these  equations — 
an  assumption  which  involves  no  appreciable  error — we  shall  have  six  equa- 
tions between  those  quantities  and  w1 '.  w,  the  longitude  of  Greenwich,  will 
be  zero. 

It  is  evident,  however,  that  for  various  reasons  a  direct  solution  of  these 
equations  will  not  be  expedient.  In  the  first  place,  the  large  terms  involved 
would  render  the  operation  very  laborious,  and  further  it  will  not  be  possible 
to  separate  Ait  from  the  remaining  quantities  without  assuming  both  w  and  iv1 
to  be  known. 

We  therefore  proceed  as  follows:  Assuming  the  equations  to  be  of  equal 
weight,  we  subtract  the  first  from  the  third,  the  first  from  the  fifth,  and  the 
third  from  the  fifth;  then  we  subtract  the  second  from  the  fourth,  the  second 
from  the  sixth,  and  the  fourth  from  the  sixth.  We  then  have  the  following  six 
equations: 


.     (B) 


By  means  of  these  six  equations  of  condition  we  now  determine  the  most 
probable  values  of  3  and  itAk.  The  value  oiArc,  however,  cannot  be  well  de- 
termined, as  we  have  before  remarked.  If  it  were  not  known  a/writhat  such 
was  the  case,  it  would  be  shown  from  the  normal  equations,  which  would  be 
practically  indeterminate  for  this  quantity.  We  shall  therefore  determine  3 
and  TtAk  in  terms  of  Ait  in  order  to  show  what  effect  an  error  in  7t  win1  have 
upon  the  longitude. 

By  the  method  of  Art.  21  we  derive  from  the  above  equations  the  following 
two  normal  equations: 

11.00563  —  5.35457TZ//&  —  —  16.0306  +  5.9864^77-;  )  ^     ^ 

—  5-3545^  -f  4.25747rJ/&  =         8.2287  —  2.8656^^.  f 


O 

o 
o 
o 
o 
o 

=       4.12 

=    3.63 

=  -    -49 

=  -f-      .12 
=   -   2.47 
=   -  2.59 

+ 
+ 

2 
I 

I. 

•337$ 
,6073 
,7303 
.9643 
.146^ 
uo3 

+ 

+ 

i.6i37fJ£ 

.^O^TtAk 
.OOOTlAk 

+ 

-f- 

+ 

I.24I2/7T; 

.2$bA7t\ 

.985/^^1 
.514^^; 

.I28J7T; 
.642Z/7T. 

[2]-[l] 

[3]-[i] 
[3]  -[2] 
[5]  -[4] 
[6]  -[4] 
[6]  -[5] 

§  268.  LONGITUDE  BY  OCCULTATION&.  473 

From  which  itAk  =          ".2588  +  .0289^/^1  )  ,^ 

-  ) 


$  =  -  i"-330i  4-  . 

We  now  substitute  these  values  in  the  first,  third,  and  fifth  of  equations  (A), 
writing  zero  for  w,  the  longitude  of  Greenwich,  when  we  find  the  following 
values  for  1.6487: 

1.6487  =  —  8.645  4~  I-2O9//7T;  \ 

1.6487  —  —  8.055  4-  i.226^jr;  !•     ........     (E) 

1.6487   =   -  5-955  +  I-276//7T.  ) 

Mean  1.6487  =  —  7.552  4-  1.237^;         y  —   -  4".  582  +  .751^. 

We  now  substitute  these  values  of  itAk,  3,  and  y  in  the  second,  fourth,  and 
sixth  of  (A),  when  we  find  the  following  values  for  the  difference  of  longitude 
between  Greenwich  and  the  observatory  on  Capitol  Hill,  Washington: 


Celaeno  w'  =  5h  8m  38.42  — 
Taygeta  w  =5  8  2  .33  — 
Maja  w'  —  5  8  i  .14  —  1.665^. 

Mean    w'  =  5    8    2  .30  —  I.686/47T. 

The  Capitol  Hill  observatory  is  io8.25  east  of  the  Naval  Observatory.  The 
longitude  of  the  latter,  determined  telegraphically,  Is  5h  8m  I28.O9  west  of  Green- 
wich. Therefore  the  true  value  of  w  is  5h  8m  is.84,  corresponding  very  closely 
with  the  above  value  if  we  neglect  ATI  altogether. 

With  these  values  of  y  and  5  we  may  now  determine  the  correction  to  the 
assumed  right  ascension  and  declination  of  the  moon. 


We  have  sin  A^cos  DA(A  —  a)  +  cos  NA(D  —  5)  =  y;  )  (      . 

-  cos  N  cos  DA(A  -  a)  -|-  sin  NA(D  -  6)  =  5  .  f 

Substituting  for  the  coefficients  of  A(A  —  a)  and  A(D  —  d)  the  mean  of  the 
values  for  the  three  stars,  we  have  the  equations 


-  a)  4-  2^A(D  -d)  =  -  4582; 

-  a)  -j-  965^/00  -  d)  =  -  1330. 

From  which  we  find  A(A  —  a)  —  —  4".  46; 

J(/>-«)  =  _2  .49. 

Assuming  the  errors  of  the  star  places  to  be  inappreciable,  these  will  represent 
the  errors  in  the  computed  right  ascension  and  declination  of  the  moon  at  a 
time  corresponding  to  the  mean  of  the  times  of  observation.  These  corrections 


4/4  PRACTICAL  ASTRONOMY.  §  269. 

it  will  be  seen  are  affected  by  any  small  outstanding  error  in  the  parallax,  as 
they  have  been  derived  by  assuming  ATI  =  o. 

'In  the  same  way,  assuming  Ait  —  o  and  taking  for  it  the  mean  of  the  values 
given  above,  viz.,  3608",  we  rind  from  the  above  value  of  TtAk 

Ak  =  -f  .0000717. 

We  have  assumed  k  =       .272270. 

Therefore  k  =       .272342, 

as  shown  from  these  observations.  This  result  from  so  small  a  number  of 
occultations  has  no  value,  however,  as  a  determination  of  the  moon's  semi- 
diarneter. 


Observations  of  Different   Weights. 

269.  In  the  solution  of  our  equations  we  have  supposed  all 
to  be  of  the  same  weight.  Such  will  not  in  general  be  the 
case.  Other  things  bejng  equal,  those  occultations  will  be 
best  for  longitude  determination,  which  are  most  nearly 
central  in  reference  to  the  moon's  disk.  When  both  immer- 
sion and  emersion  oi  the  same  star  are  observed,  the  obser- 
vation at  the  dark  limb  of  the  moon  is  entitled  to  greater 
weight  than  that  at  the  bright  limb,  except,  perhaps,  in  case 
of  the  brighter  stars. 

In  order  to  determine  the  proper  manner  of  treating  the 
equations  when  different  weights  are  assigned,  let  us  suppose, 
as  in  our  example,  three  observations  to  have  been  made  at 
one  place  whose  true  longitude  is  w,  then  for  the  present, 
considering  only  terms  in  y  and  5,  we  shall  have  three  equa- 
tions of  this  form  : 

Vpw  Vpa$         -  VpO     =  o; 

Vp'w   -       Vp'a'5       •  Vp'O'  =  o;     ['  :     .     (4.66) 

Vp^w  -     Vp^a"^  -  ^0"=  o. 


§  269.  LONGITUDE  BY  OCCULTATIONS.  4?$ 

Where  O  —  /  1  —  vy,  and/,/',/"  are  the  respective  weights. 
From  these  we  derive  the  normal  equations 


=  o;l 

=  o.  ) 


The  solution  of  these  equations  in  the  usual  manner  gives 


[paai\  $  =  [paOi]. 

Which  gives  $  with  the  weight  [paai]. 

But,  as  we  have  seen,  this  form  of  solution  is  inconvenient 
on  account  of  the  large  quantities  involved. 

Let  us  write  out  in  full  the  values  of  \_paai~]  and  [paOi]: 


[paai\  =p 

_  pp'(a  -  aj  +pp"(a  -  a'J  -\-p'p"(a'  -  a" 


P+P'  +P"  '  J 

Comparing  these  expressions  with  our  equations  of  condi- 
tion (466),  we  see  that  the  final  equation  for  5  may  be 
obtained  as  follows:  Before  multiplying  the  equations 
through  by  Vp,  Vp\  and  Vp",  subtract  the  second  from 
the  first,  the  third  from  the  first,  and  the  third  from  the 


PRACTICAL   ASTRONOMY.  §  269. 

second,  then  give  to  the  three  resulting  equations  the  fol- 
lowing weights  respectively: 

pp'  pp"  p'p" 

PP  PP  PP  (4;o) 


We  may  apply  the  same  reasoning  to  the  equation  in  which 
all  of  the  unknown  quantities  are  retained,  and  may  extend 
it  to  any  number  of  equations  of  condition.  Thus  if  the 
number  of  equations  of  condition  were  four,  we  find  by  com- 
bining them  in  a  like  manner,  two  and  two,  six  equations 
wifh  weights 

PP'  PP"  / 


_          _ 
/  +/  +/'  +/""/  +/  +/'  +/"         '  P  +/'  +>"  +P"" 

It  is  not  possible  to  give  a  rule  by  which  the  proper 
weight  can  be  assigned  in  every  case,  as  it  will  depend  upon 
a  variety  of  circumstances,  such  as  the  skill  and  experience 
of  the  observer,  the  magnitude  of  the  star,  condition  of  the 
atmosphere,  and  various  other  causes.  Evidently,  if  weights 
are  to  be  assigned  depending  upon  these  circumstances, 
much  must  be  left  to  the  judgment  of  the  observer  and  com- 
puter. If  the  conditions  are  otherwise  the  same  in  case  of 
two  stars,  the  weights  may  be  assumed  proportional  to  the 
numerical  values  of  cos  ip  ;  that  is,  proportional  to  the  chord 
of  the  moon's  disk  traversed  by  the  stars  —  a  central  occulta- 
tion  having  the  weight  unity. 

If  we  assign  weights  to  our  six  equations  (A)  in  accordance  with  this  prin- 
ciple, we  shall  have  for  the  weights,  taken  in  order,/  =  .49;  /i  =  i.oo;  /'  =.94; 
pi  =  .84;  p"  =  .58;  pi"  —  i.oo. 

The  weights  of  equations  (B)  will  then  be  in  accordance  with  formulae  (470). 


[2]  -  [I]   .229  [5]  -  [4]   .296 

[3]  -  [i]  .141         [6]  -  [41  .352 
[3]  ~  [2]  -271         [6]  -  [5]  -296 


§269.  LONGITUDE  BY  OCC  ULTATIONS.  477 

Multiplying  the  equations   by  the   square  roots  of  the  respective  weights  and 
proceeding  in  the  usual  way,  we  obtain  the  following  normal  equations: 


2.7630^  —  i.239i7rJ>£  =  —  3.7605  -|-  1.5129^/7?; 
—  1.23913  -j-  i  oi747T^^  —        1.6907  —    .6678//7T. 


From  these^we  find  TtAk  —  -f    .00931  -\-  .0232^; 

3  =  -  1.3570    +  .5579^*- 

Substituting  these  values  in  [i],  [2],  and  [3]  of  equations  (A),  and  taking  the 
mean  by  weights,  we  find 

1.6487   =    —  8.l6l  -f-  I.22I^7T. 

Finally,   substituting  these  values  of  3,   TtAk,   and  y  in  [4],  [5],  and  [6],  we 
find  the  following  values  for  w  '  : 

[4]    w   =  5h  8m  3s.  61  —  1.706^/71:;    wt.  =  i.oo. 
[5]    w'  =  $    8    2  .43  —  1.675^;   wt.  =    .84^ 
[6]    w   =  5    8    i  .34  —  i.  66oJit;   wt.  =  i.oo. 

From  these  we  have 

•w  =  sh  8m  28.46  — 


CHAPTER  VIII. 

•THE  ZENITH  TELESCOPE. 

270.  This  instrument  is  used  in  determining  latitude,  and 
is  particularly  useful  when  a  high  degree  of  accuracy  is  re- 
quired, the  precision  being  not  inferior  to  that  of  the  most 
refined  instruments  of  a  fixed  observatory,  while  on  account 
of  its  great  simplicity  it  is  especially  adapted  to  use  in  the 
field.  * 

We  have  already  developed  several  methods  for  determin- 
ing latitude :  those  of  Chapter  V.  are  very  useful,  but  will 
not  be  employed  in  the  field  except  in  cases  where  an  error 
of  five  or  six  seconds  in  the  result  is  not  considered  objec- 
tionable. The  prime  vertical  transit  gives  results  of  high 
precision,  but  not  without  the  expenditure  of  much  labor. 
The  method  by  the  zenith  telescope  is  superior  to  the  first 
of  these  in  accuracy,  and  to  the  second  in  facility  of  applica- 
tion. On  account  of  these  advantages  it  has  superseded  all 
other  methods  on  the  Coast  and  other  government  surveys 
in  cases  where  extreme  accuracy  is  required. 

The  most  common  form  of  instrument  is  shown  in  Fig.  54. 
In  general  appearance,  as  will  be  seen,  it  is  a  telescope  with 
an  altitude  and  azimuth  mounting.  The  essential  character- 
istics are  a  very  delicate  level  attached  to  the  tube,  like  the 
level  of  the  finding-circles  in  the  transit  instrument,  and  the 
eye-piece  micrometer.  The  vertical  axis  is  made  very  long 
to  insure  steadiness  of  motion  in  azimuth.  The  instrument 
is  used  in  the  meridian  like  the  transit. 


§2/0. 


THE   ZENITH   TELESCOPE. 


479 


FIG.  54.— THE  ZENITH  TELESCOPE. 


4^0  l^RACTICAL   ASTRONOMY.  §  2 70. 

In  the  Coast  Survey  instrument  the  aperture  of  the  tele- 
scope is  3^  inches,  focal  length  45  inches,  length  of  horizon- 
tal axis  7  inches,  vertical  axis  24  inches,  diameter  of  horizon- 
tal circle  12  inches,  vertical  circle  6  inches  (sometimes  this 
is  only  a  semicircle,  the  radius  being  6  inches).  -The  instru- 
ment rests  on  three  foot-screws.  The  lamp  at  the  end  of  the 
horizontal  axis  opposite  the  telescope  illuminates  the  field ; 
the  weight  seen  at  the  same  end  of  the  axis  acts  as  a  counter- 
poise to  the  telescope.  This  weight  is  connected  with  the 
telescope  by  a  bent  metallic  bar,  shown  in  the  figure,  in  such 
a  way  as  to  prevent  to  some  extent  the  flexure  of  the  axis. 

The  horizontal  circle  is  read  by  means  of  two  verniers. 
The  level  attached  to  the  vertical  circle  is  generally  gradu- 
ated so  that  the  motion  of  the  bubble  over  one  millimetre 
corresponds  to  an  angle  of  one  second  of  arc.  The  accuracy 
of  the  instrument  depends  in  a  great  degree  on  the  delicacy 
of  this  level.  In  testing  an  instrument  it  may  generally  be 
assumed  that  if  the  level  is  a  good  one  the  performance  of 
the  instrument  as  a  whole  will  be  satisfactory.  The  striding- 
level  shown  on  the  horizontal  axis  is  used  for  adjusting  the 
instrument,  and  is  not  necessarily  of  so  great  accuracy. 

The  micrometer*  is  provided  with  one  or  more  movable 
threads,  the  value  of  one  revolution  of  the  screw  being  from 
45"  to  60".  The  head  of  the  screw  is  divided  into  100  parts, 
of  which  tenths  may  be  estimated;  thus  by  estimation  ToVo  °f 
one  revolution  may  be  read,  or  about  o".o$.  The  entire 
revolutions  are  read  by  means  of  a  comb  at  one  side  of  the 
field  of  view,  the  distance  between  two  consecutive  notches 
corresponding  to  one  revolution.  There  are  three,  and 
sometimes  five,  vertical  threads  which  may  be  used  for 
observing  transits.  A  rack  and  pinion  is  provided  for  slid- 
ing the  eye-piece  in  the  direction  of  the  vertical  so  that  the 
star  may  always  be  observed  in  the  middle  of  the  field. 

*  For  description  of  the  micrometer  see  Art.  97. 


§2/1.  .ZENITH   TELESCOPE.— ADJUSTMENTS.  481 

The  instrument  is  mounted  like  the  transit  on  a  pier  of 
masonry,  or  simply  a  solid  wooden  post  planted  three  feet  in 
the  ground. 

The  dimensions  given  above  are  those  of  a  large-sized  in- 
strument; much  smaller  ones  are  often  used. 

The  transit  instrument  may  be  used  as  a  zenith  telescope 
if  it  is  provided  with  the  fine  level  and  micrometer.  A 
special  appliance  for  reversing  is  convenient,  but  not  essential. 
As  we  have  seen  in  the  descriptions  of  the  different  forms  of 
portable  transit  instruments,  the  two  are  often  combined. 
This  arrangement  is  very  advantageous  on  the  ground  of 
economy  of  first  cost  and  of  transportation;  at  the  same  time 
nothing  is  lost  in  accuracy  and  little  in  convenience. 


Adjustments. 

271.  First.  The  vertical  axis  must  be  made  truly  vertical. 
In  setting  up  the  instrument  it  will  be  found  advisable  to 
place  two  of  the  foot-screws  in  an  east  and  west  direction, 
otherwise  if  it  is  found  necessary  to  move  the  screws  after 
the  instrument  has  been  brought  into  the  plane  of  the  me- 
ridian this  last  adjustment  will  be  disturbed. 

The  axis  is  brought  into  the  vertical  position  by  the  use 
of  the  strid  ing-lev  el,  which  should  read  the  same  while  the 
instrument  is  turned  completely  around  in  azimuth.  This 
adjustment  will  also  be  tested  by  means  of  the  more  delicate 
level  attached  to  the  telescope. 

Second.  The  horizontal  axis  should  be  perpendicular  to  the 
vertical  axis.  This  may  be  tested  by  reversing  the  striding- 
level  after  the  vertical  axis  has  been  properly  adjusted. 

Third.  The  line  of  collimation  may  be  adjusted  by  direct- 
ing the  telescope  to  some  distant  terrestrial  mark,  then  turn- 
ing the  instrument  180°  in  azimuth  by  means  of  the  horizontal 


4^2  PRACTICAL   ASTRONOMY.  §  271. 

circle.  Allowance  must  be  made  for  the  parallax  of  the  in- 
strument,  unless  the  mark  is  so  far  away  that  it  is  not  appre- 
ciable. This  is  necessary,  since  the  line  of  collimation  is  not 
in  the  same  vertical  plane  as  the  axis. 

Let  d  =  distance  of  the  line  of  collimation  from  vertical 

axis; 

D  =  distance  of  mark ; 
p  =  correction  for  parallax. 

Then  f  =  TOT* (470 

This  method  of  adjustment  depends  entirely  on  the  read- 
ing of  the  circle,  and  is  therefore  not  capable  of  extreme  ac- 
curacy. If  considered  desirable,  a  more  accurate  adjustment 
may  be  made  by  means  of  a  pair  of  collimating  telescopes* 
or  by  the  mercury  collimator.*  The  error  may  also  be  de- 
termined by  transits  of  stars  observed  in  both  positions  of 
the  axis,  as  explained  in  connection  with  the  transit  instru- 
ment. If  stars  are  chosen  which  culminate  near  the  zenith, 
an  error  of  azimuth  will  have  but  little  influence  on  the  re- 
sult. 

When  used  as  a  transit  instrument  a  meridian  mark  is 
recommended,  consisting  of  two  lamps  placed  side  by  side 
and  at  a  distance  apart  equal  to  twice  the  distance  of  the 
vertical  from  the  collimation  axis. 

It  is  perhaps  unnecessary  to  say  that  the  instrument  must 
be  focused  and  the  threads  placed  truly  vertical  and  hori- 
zontal respectively,  precisely  as  in  the  transit  instrument. 

Fourth.  The  instrument  must  be  brought  into  the  plane  of 
the  meridian.  For  this  and  other  purposes  we  require  the 
local  time,  a  chronometer  or  clock  being  an  essential  part  of 

*  See  Art.  168. 


§  2/1.  ZENITH   TELESCOPE.— ADJUSTMENTS.  483 

the  outfit.  The  clock  correction  A  T  may  be  determined  by 
the  sextant,  transit  instrument,  or  by  transits  observed  with 
the  zenith  telescope  itself.  In  the  latter  case  the  process  of 
bringing  the  instrument  into  the  meridian  will  be  the  same 
as  that  already  described  for  the  transit. 

If  A  T  is  known  within  one  second  of  its  true  value,'  that 
will  be  sufficient. 

AT  being  supposed  known, 

Let  OL  =  the  right  ascension  of  a  star  near  the  pole. 

Then  a  —  AT  =  the  chronometer  time  of  culmination. 

At  this  instant,  as  shown  by  the  chronometer,  the  middle 
thread  is  placed  on  the  star,  the  horizontal  circle  being  pro- 
vided with  a  clamp  and  tangent-screw  for  this  and  similar 
purposes.  The  reading  of  the  verniers  now  shows  the  true 
direction  of  the  meridian.  Two  stops  arranged  for  the  pur- 
pose are  now  clamped  to  the  horizontal  circle  so  that  the  in- 
strument may  be  turned  freely  in  azimuth,  but  brought  to  a 
stop  when  it  reaches  the  meridian.  Care  must  be  taken  in 
turning  the  instrument  in  azimuth  not  to  bring  it  up  against 
these  stops  with  a  shock,  as  this  will  disturb  the  adjustment. 

South  stars  may  be  used  for  adjusting  in  the  meridian,  pro- 
vided they  are  sufficiently  far  from  the  zenith.  In  any  case 
the  adjustment  should  be  tested  by  trying  whether  a  south 
star  crosses  the  middle  thread  at  the  proper  time. 

The  stops  should  be  placed  so  that  in  reversing  the  in- 
strument in  azimuth  the  object  end  of  the  telescope  always 
turns  towards  the  east.  The  observer  can  then  turn  it  in 
azimuth  a  little,  so  as  to  find  a  star  a  moment  before  it  enters 
the  field  ;  then  knowing  exactly  where  to  look  for  the  star, 
the  eye-piece  can  be  brought  to  the  right  place  by  the  rack 
and  pinion,  and  the  micrometer-thread  moved  to  nearly  the 
proper  place,  so  that  when  the  star  finally  comes  into  view 
the  bisection  can  be  made  with  all  necessary  deliberation. 


484  PRACTICAL   ASTRONOMY.  §  272 

All  of  the  above  matters  having  been  attended  to,  the  in- 
strument is  ready  for  regular  latitude  observation. 


The  Observing  List. 

272.  The  stars  are  observed  in  pairs,  one  star  culminating 
north  of  the  zenith  and  the  other  south.  The  difference  of 
zenith  distance  should  not  exceed  15'  or  20'. 

Let  <p,  tf,  and  6'  =  respectively  the  latitude  of  station  and 

declination  of  south  and  north  star ; 
z  and  z'  —  the  zenith  distances. 

Then  cp  =  3  +  z\ 

cp  =  6'  -  z'-, 

*>  =  *(*  +  *')  +  *(*-*') (472) 

Thus  the  latitude  is  equal  to  one  half  the  sum  of  the  declina- 
tions plus  one  half  the  difference  of  zenith  distance,  which 
latter  must  be  small  enough  to  be  capable  of  measurement 
by  the  micrometer. 

The  difference  of  right  ascension  of  the  two  stars  forming 
the  pair  should  not  exceed  I5m  or  2Om,  as  changes  may  take 
place  in  the  instrument  if  a  longer  time  elapses.  If  care  is 
used  in  the  selection,  it  will  seldom  be  necessary  to  use  a 
pair  with  so  long  an  ^interval  as  15  minutes.  The  interval 
should  not  be  less  than  one  minute,  as  the  instrument  must 
be  read  and  reversed  in  azimuth  for  the  second  star,  which 
will  require  at  least  that  amount  of  time. 

Stars  smaller  than  the  /th  magnitude  cannot  be  well  ob- 
served with  the  instrument  which  has  been  described.  With 
smaller  instruments  the  6th  magnitude  will  be  about  the 
limit. 


§  2/2.  OBSERVING  LIST.  485 

Stars  at  any  zenith  distance  may  be  observed,  but  gen- 
erally it  will  not  be  necessary  or  advisable  to  go  beyond  30° 
or  35°. 

The  catalogues  most  suitable  for  the  selection  of  stars  are 
the  Coast  Survey  catalogue,*  the  various  Greenwich  cata- 
logues, and  the  British  Association  catalogue.  The  declina- 
tions of  the  latter  are  not  sufficiently  reliable  for  a  good  lati- 
tude determination;  but  as  it  contains  nearly  all  the  stars  down 
to  the  6th  magnitude  inclusive,  it  may  very  conveniently  be 
used  in  selecting  the  list,  the  final  declinations  being  after- 
wards taken  from  more  reliable  catalogues. 

In  selecting  the  stars  we  require  an  approximate  value  of 
the  latitude,  which  may  often  be  taken  from  a  map  with  suf- 
ficient accuracy,  or  if  suitable  maps  are  not  available  it  may 
be  determined  by  a  single  altitude  of  the  sun  or  a  star  at  cul- 
mination measured  with  the  sextant.  An  error  of  if  or  2'  in 
the  assumed  value  will  cause  no  inconvenience. 

In  selecting  the  list  of  stars  we  proceed  as  follows  :  First 
we  must  know  with  what  right  ascension  to  begin.  If,  for  in- 
stance, we  intend  beginning  our  observations  at  7h  P.M.,  this 
mean  solar  time  converted  into  sidereal  time  will  give  the 
right  ascension  of  a  star  which  culminates  at  that  instant. 
Starting  with  this  right  ascension,  we  take  the  first  star 
whose  zenith  distance  at  culmination  does  not  exceed  35°  and 
look  down  the  list  to  find  whether  there  is  another  star  which 
differs  from  this  in  right  ascension  between  im  and  15™,  and 
which  will  unite  with  this  to  form  a  suitable  pair.  From 
(472)  we  have 

=  2?  -V  -(*-*>);) 


Thus  if  d'  is  the  declination  of  the  star,  if  we  can  find  another 

*  Coast  Survey  Report  1876,  Appendix  No.  7. 


486  PRACTICAL  ASTRONOMY.  §  272, 

whose  declination  d  does  not  differ  from  2cp  —  d'  more  than 
15'  or  20',  the  two  stars  will  form  a  pair  suitable  for  our  pur- 
pose. With  the  great  majority  of  trials  we  shall  find  no 
second  star  fulfilling-  the  above  conditions.  If  we  use  the 
British  Association  catalogue  we  can  generally  find  from 
one  to  three  dozen  pairs  suitable  for  observation  for  any 
night  in  the  year. 

Having  gone  over  the  catalogue  in  this  manner,  writing 
down  the  catalogue  numbers  of  the  stars,  the  right  ascen- 
sions, declinations,  and  magnitudes,  it  will  often  be  found 
that  some  of  the  pairs  interfere  with  others  in  reference  to 
time  of  culmination.  We  may,  if  we  choose,  make  out  two 
lists  for  observation  on  alternate  nights,  or  we  may  drop 
those  pairs  which  are  less  suitable  when  they  interfere  with 
others. 

The  places  of  the  stars  must  then  be  reduced  to  the  date 
of  observation  by  applying  the  corrections  for  precession, 
nutation,  and  aberration.*  The  declinations  need  only  be 
reduced  to  the  mean  place  for  the  year,  but  the  apparent 
right  ascensions  for  the  date  of  observation  will  be  required 
within  the  nearest  second.  The  necessary  reduction  may  be 
obtained  very  readily  by  comparing  the  stars  with  those  of 
approximately  the  same  right  ascension  and  declination  of 
the  Nautical  Almanac. 

The  following  is  an  example  of  an  observing  list  prepared 
for  determining  the  latitude  along  the  northern  boundary  of 
the  United  States.  The  first  column  contains  the  number 
of  the  star  in  the  British  Association  catalogue,  the  second 
column  the  magnitude,  the  third  and  fourth  the  right  ascen- 
sion and  declination,  the  fifth  the  zenith  distance.  The  let- 
ter N.  or  S.  in  the  next  column  shows  whether  the  star  cul- 
minates north  or  south  of  the  zenith  :  the  stars  with  the  large 

*For  a  full  explanation  of  this  subject  see  Art.  354  and  following. 


§2/3. 


OBSERVING  LIST. 


487 


declinations  culminate  north,  those  with  the  small  declina-    . 
tion  south.    The  setting,  given  in  the  last  column,  is  the  mean 
of  the  zenith  distances.  t 

U.  S.  Northern  Boundary  Survey. — Astronomical  Station  No.  4. 
Observing  List  for  Zenith  Telescope.  1873,  June  27.         Approx.  cp  49°  o'. 


B.  A.  C. 

Mag. 

a 

8 

z 

N.  or  S. 

Setting. 

4937 
4974 

6 

5 

I4h  52m  12s 

14  59  38 

.  50°  9' 

48  9 

J°  9' 

o  51 

N. 
S. 

1°  00' 

5026 
5097 

6 

3 

15  8  47 

15  22    8 

38  44 
59  24 

10  16 
10  24 

S. 

N. 

10  20 

5271 
5313 

6 

5-5 

15  48  19 

15  54  49 

42  48 
55   7 

6  12 
6   7 

S. 

N. 

6  9-5 

5415 
5460 

6 
6 

16  6  36 
16  15  36 

58  16 
40   i 

9  16 
8  59 

N. 
S. 

9  7-5 

5502 

5523 

5 
5 

16  21  41 
16  24  31 

55  30 
42  10 

6  30 
6  50 

N. 
S. 

6  40 

5545 
5624 

4-5 

7 

16  28  17 
16  40   4 

69   3 

28  35 

20   3 
20  25 

N. 
S. 

20  14 

5644 
5658 

6 
6 

16  43  18 
16  44  17 

42  28 
55  38 

6  32 
6  38 

S. 

N. 

6  35 

As  will  be  seen,  the  selection  of  a  good  list  of  stars  involves 
considerable  labor.  Where  great  accuracy  is  required 
especial  care  should  be  exercised  in  selecting  the  stars,  and 
none  should  be  employed  whose  declinations  are  not  well 
determined.  This  part  of  the  subject  will  be  considered 
more  in  detail  hereafter. 

Directions  for  Observing. 

273.  A  suitable  list  of  stars  having  been  prepared,  the  in- 
strument adjusted,  and  the  chronometer  error  determined, 
the  observer  sets  the  vertical  circle  at  the  proper  reading, 
the  telescope  is  directed  towards  that  side  of  the  zenith 


4-S8  PRACTICAL   ASTRONOMY.  §  274. 

where  the  first  star  will  culminate,  and  the  bubble  brought 
to  the  middle  of  the  level-tube  by  means  of  the  tangent- 
screw  connected  with  the  horizontal  axis.  At  the  time 
of  culmination,  as  shown  by  the  chronometer,  the  star  is  bi- 
sected by  the  micrometer-thread,  and  the  micrometer  and 
level  are  read ;  the  instrument  is  then  reversed  in  azimuth 
and  the  second  star  observed  in  the  same  way :  this  forms  a 
complete  observation. 

During  the  operations  described  the  tangent-screw  of 
the  vertical  circle  must  not  be  touched,  but  the  tangent- 
screw  which  moves  the  telescope,  and  consequently  the  level, 
may  be  turned  after  reversing,  in  the  exceptional  case  where 
the  vertical  axis  is  not  well  adjusted. 

If  for  any  reason  the  bisection  is  not  obtained  at  the  in- 
stant of  culmination,  the  star  may  be  observed  off  the  meridian 
and  the  time  of  observation  recorded,  when  a  correction  may 
be  computed  to  reduce  it  to  the  meridian.  Several  bisections 
might  be  made  while  the  star  is  crossing  the  field,  and  the 
observations  reduced  to  the  meridian  in  a  similar  manner  ; 
but  experience  shows  that  little  or  nothing  is  gained  in  this 
way.  The  accuracy  with  which  a  bisection  can  be  made  by 
a  skilled  observer  being  greater  than  that  of  the  average  de- 
clinations which  will  be  employed,  it  is  advisable  to  increase 
the  number  of  stars  observed  rather  than  to  multiply  obser- 
vations on  the  same  star  under  the  same  circumstances. 


Determination  of  Value  of  Micrometer-screw. 

274.  This  value  may  be  determined  most  advantageously 
by  means  of  a  circumpolar  star  observed  near  elongation. 
One  of  the  four  close  circumpolar  stars  whose  places  are 
given  in  the  American  Ephemeris  will  generally  be  selected 
for  the  purpose,  viz.,  51  Cephei,  $,  a,  or  A  Ursae  Minoris. 


§2/4-       DETERMINATION  OF  MICROMETER    VALUE. 


489 


The  observations  are  made  as  follows:  From  15  to  30  min- 
utes before  the  star  reaches  elongation  the  telescope  is 
pointed  to  the  star,  the  micrometer-thread  being  near  that 
end  of  the  screw  from  which  the  star  is  moving.  The  tele- 
scope is  set  at  such  an  elevation  that  the  thread  is  a  little  in 
advance  of  the  star,  and  the  bubble  of  the  level  brought  into 
the  middle  of  the  tube,  without  disturbing  the  position  of 
the  telescope.  The  time  of  transit  of  the  star  over  the  thread 
is  then  observed  and  the  level  read.  The  thread  is  then 
moved  forward  one  revolution  (or  sometimes  only  half  a 
revolution)  and  the  transit  of  the  star  observed  in  the  new 
position,  and  so  on  throughout  the  entire  length  of  the 
screw. 

It  is  well  to  time  the  work  $o  that  the  elongation  will 
occur  near  the  middle  of  the  series,  though  this  is  not  essen- 
tial. With  this  in  view  it  may  be  borne  in  mind  that  the 
time  required  for  Polaris  to  pass  over  a  space  equal  to  the 
range  of  an  ordinary  zenith  telescope  micrometer  will  be 
about  5om,  for  A.  Ursas  Minoris  70™,  for  51  Cephei  30™. 

The  record  of  the  observations  will  be  kept  according  to 
the  following  or  a  similar  schedule  : 


No. 

Micrometer. 

Chronom.  Time. 

Level. 

N. 

S. 

To  prepare  for  the  observation,  the  chronometer  time  of 
elongation  must  be  computed.  It  will  facilitate  setting  the 
instrument  on  the  star  if  the  azimuth  and  zenith  distance  are 
also  computed. 


49°  PRACTICAL  ASTRONOMY.  §  275- 

In  the  triangle  formed  by  the  arcs  of  great  circles  joining 
P    9o°-&  s  the  zenith,  the  pole,  and  the  star,  the  angle  at  the 
star  5  will  be  a  right  angle  at  the  time  of  elonga- 
tion.    Then  by  Napier's  rules, 

'z  cos  d 

sin  a  = 


cos 
sin  < 


cos  z  =  -- 


(474) 


sm  6  ' 

cos  /  =  tan  (p  cot 
Let  T  =  the  chronometer  time  of  elongation. 

Then  T  = 'a  ±  t  —  dT\  T-  '  >  elongation.    .     .    (474^ 

Method  of  Reduction. 

275.  We  have  by  observation  a  series  of  times  correspond- 
ing to  observed  transits  of  the  star  over  the  thread  at  succes- 
sive equal  distances.  If  now  the  star  moved  uniformly  in  a 
great  circle  the  intervals  between  these  observed  times  would 
be  .uniform,  aside  from  errors  of  observation  and  the  effect 
of  change  of  level.  The  star,  however,  moves  in  a  small 
circle  which  is  tangent  to  the  vertical  circle  at  the  point  of 
elongation.  We  may,  however,  compute  the  correction 
necessary  to  convert  this  motion  in  the  small  circle  to  uni- 
form motion  in  a  great  circle,  as  follows: 

For  any  one  of  our  observed  transits  let 
t    =  the  interval  of  time  between  ob- 
servation and  elongation  ; 
z"  =  the  number  of  seconds  of  arc  from 

elongation     measured     on    the  FIG.  5e. 

vertical  circle  =  SK. 
Then  the  angle  SPK  =15^  expressed  in  arc,  and 

sin  *"  =  cos  8  sin  (IJT) (475) 

.  sin  (157) 

or  *"  =  cos  d  -..:  ,,/ 


§2/5-       DETERMINATION  OF  MICROMETER    VALUE.  491 


By  expansion, 

sin(isr)  =  (isr)sin  i"-|(i5rsiii 


sn 


If  the  time  of  elongation  falls  anywhere  within  the  series 
the  last  term  is  never  likely  to  be  appreciable,  so  we  shall 
have  with  sufficient  accuracy 


z"  =  15  cos  d  [r  -  £(15  sin  i")V]. 
In  which         log  £(15  sin  i")2  =  0.94518  —  10. 


(475\ 


This  term  may  be  readily  computed  from  the  formula,  but 
the  following  table  is  more  convenient,  where  its  value  is 
given  for  every  minute  of  time  from  elongation  to  65™.  It 
will  seldom  be  advisable  to  extend  the  observations  farther 
from  elongation  than  this.  For  this  interval,  viz.,  65m,  the 
term  in  rb  is  os.2i,  and  may  very  well  be  neglected,  but  it 
would  soon  become  appreciable. 


T 

Term. 

r 

Term. 

T 

Term. 

m. 

s. 

,  m. 

s. 

m. 

j. 

6 

o.o 

26 

3-3 

46 

18.5 

7 

O.I 

27 

3-7 

47 

19.7 

8 

O.I 

28 

4.2 

48 

21.  0 

9 

O.I 

29 

4-6 

49 

22.3 

10 

0.2 

30 

5-i 

50 

23-7 

ii 

O.2   . 

3< 

5-7 

51 

25.2 

12 

0-3 

32 

6.2 

52 

26.7 

-T3 

0.4 

33 

6.8 

53 

28.3 

14 

0.5 

34 

7-5 

54 

29.9 

15 

0.6 

35 

8.2 

55 

31-6 

16 

0.8 

36 

8.9 

56 

33-3 

17 

0.9 

37 

9.6 

57 

35-1 

18 

i.i 

38 

10.4 

58 

37-0 

19 

i-3 

39 

H-3 

59 

39-° 

20 

1-3 

40 

12.2 

60 

41.0 

21 

1.8 

41 

I3-I 

61 

43-i 

22 

2.0 

42 

I4.I 

62 

45-2 

23 

2-3 

43 

I5-I 

63 

47-4 

24 

2.6 

44 

16.2 

64 

49  7 

25 

3-0 

45 

17.3 

65 

52.1 

492  PRACTICAL  ASTRONOMY.  §  2/6. 

Instead  of  applying  this  correction  to  r  (the  difference  be- 
tween  the  time  of  elongation  and  observation)  it  is  more  con- 
venient to  apply  it  directly  to  the  observed  time.  It  will  be 
plus  before  and  minus  after  either  elongation.  We  thus 
reduce  the  observed  times  to  what  they  would  have  been  if 
the  star  had  moved  uniformly  in  a  vertical  circle. 

276.  Correction  for  Change  of  Level  Reading.  A  change  in 
the  level  reading  indicates  a  change  in  the  angle  which  the 
line  of  collimation  forms  with  the  horizon.  The  correction 
necessary  to  apply  to  the  observed  times  will  be  derived  as 
follows  :  . 

Let  n,  s  =  any  level  reading  ; 

«o,  s0  =  an  assumed  level  reading  to  which  all  are  to  be 
reduced. 

Then  /  =  d\&n  -  s)  -  i(»0  -  j0)]. 

This  quantity  will  be  an  increment  to  z"  ,  and  since  it  will 
always  be  very  small  it  may  be  treated  as  a  differential.  To 
find  the  necessary  correction  to  r  we  differentiate  equation 

(475): 

cos  z"  dz"  •=.  cos  S  cos  (157)  ^(15  r). 

Writing   dz"  —  /,          cos  z"  —  I,          cos  i^r  —  I, 
this  gives 

'  (476) 


Applying  this  and  the  correction  taken  from  the  table  Art. 
275  to  the  observed  times,  we  shall  have  in  one  column 
the  readings  of  the  micrometer,  and  in  another  the  times 
reduced  to  what  they  would  have  been  if  the  star  had  moved 
uniformly  in  vertical  circle,  and  if  no  change  had  taken  place 
in  the  position  of  the  instrument.  These  may  now  be  com- 


§2/7-       DETERMINATION  OF  MICROMETER    VALUE.  493 

bined  by  subtracting  the  first  from  the  middle  one,  the  sec- 
ond from  the  middle  plus  one,  and  so  on. 

If  n  is  the  number  of  revolutions  of  the  micrometer  between 
the  first  and  middle  observations,  we  thus  have  a  series  of 
values  for  the  time  required  for  the  star  to  pass  over  this 
space;  if  all  errors  could  be  avoided,  these  times  would  con- 
sequently be  the  same.  The  mean  of  these  values  multiplied 

by  -^—      — ,  in  accordance  with  formula(475)1,  then  gives  the 

value  of  one  revolution  expressed  in  seconds  of  arc. 

277-  Micrometer  Value  when  Level  Value  is  not  knoivn.  There 
is  no  more  convenient  or  satisfactory  method  for  determin- 
ing the  value  of  the  micrometer-screw  than  that  just  explained, 
when  the  value  of  the  level  has  been  previously  determined. 
This  may  be  done  by  a  level-trier,  or  by  a  finely  graduated 
circle,  as  already  explained  in  Art.  164. 

Circumstances  sometimes  make  it  necessary  to  determine 
the  values  of  both  micrometer  and  level  when  no  special  ap- 
pliances are  at  hand  for  the  latter.  In  such  a  case  the  value 
of  the  level  must'first  be  determined  in  terms  of  the  microm- 
eter, as  follows : 

The  telescope  is  directed  to  a  sharply-defined  mark,  as  the 
threads  of  a  collimating  telescope,  and  the  bubble  brought 
near  one  end  of  the  tube ;  the  mark  is  carefully  bisected  by 
the  thread  of  the  micrometer,  and  both  micrometer  and  level 
are  read.  The  instrument  is  then  moved  through  a  small 
vertical  angle  so  as  to  bring  the  bubble  towards  the  other 
end  of  the  tube,  and  the  mark  again  bisected  by  the  microm- 
eter. 

The  difference  between  the  two  readings  of  the  microm- 
eter is  the  measure  of  the  angle  through  which  the  instrument 
has  been  moved  in  terms  of  the  micrometer,  and  the  differ- 
ence between  the  two  level  readings  is  the  measure  of  the 
same  angle  in  terms  of  the  level. 


494  PRACTICAL   ASTRONOMY.  §  277. 

Let  M,  M'  =  the  two  micrometer  readings  ; 
L,  L'  —  the  two  level  readings  ; 
R,  d  —  value  of  micrometer  and  level  respectively. 

Then  d(L  -  Lf)  =  R(M  -  M'}  .....    (477) 

The  value  of  both  d  and  R  may  now  be  determined  by  a 
series  of  approximations,  as  follows  :  The  value  of  R  is  deter- 
mined by  the  method  just  explained,  neglecting  the  level 
correction  ;  then  with  this  value  of  R,  </is  computed  by  (477), 
and  the  value  used  in  a  recomputation  of  R.  This  more 
accurate  value  of  R  gives  a  more  accurate  approximation  to 
the  value  of  </,  and  the  operation  may  be  again  repeated  if 
necessary.  If  the  instrument  is'mounted  on  a  good  founda- 
tion, the  change  of  level  during  the  time  of  observation  will 
generally  be  so  small  that  a  very  close  approximation  to  the 
true  value  of  R  is  obtained  by  neglecting  the  level  correc- 
tion. It  will  seldom  happen  that  the  change  will  be  great 
enough  to  render  more  than  one  repetition  of  the  computa- 
tion necessary. 

A  method  theoretically  more  rigorous  is  as  follows  : 

Let  p=-j  --  =7-=Z>:=the  value  of  one  division  of  the 

level  expressed  in  terms  of  the 
micrometer  ; 

ZM  TOJ  M0,  L0  =  zenith  distance,  time,  micrometer, 
and  level  of  a  circumpolar  star 
observed  at  elongation  ; 
2,  T,  M,  L  —  the  same  quantities  at  time  T. 
RD  —  d  —  value  of  one  division  of  the  level 


Then  z  =  *0  +  (M   -  M0)R  -  (L   -  L^RD, 

-  M0)R  -  (Lf  - 


for  a  second  observation. 


§  2/8.       DETERMINATION  OF  MICROMETER    VALUE.  49$ 

From  these,      R  =  (j/_  M]  _  (//  _~J^    '     '     *    (477)l 


z  —  z9t  z'  —  #0  are  the  same  as  the  quantity  which  we 
have  called  z"  in  the  previous  formula,  and  may  be  com- 
puted by  (475).  The  correction  £(15  sin  i")V  may  of  course 
be  taken  from  the  table  and  applied  directly  to  the  time  of 
observation  as  before.  We  shall  then  have  in  one  column 
the  readings  of  the  micrometer,  and  in  another  the  times  re- 
duced to  the  vertical  circle.  We  combine  as  before  by  sub- 
tracting the  first  from  the  middle,  the  second  from  the 
middle  plus  one,  and  so  on;  then  divide  each  by  its  value  of 
(M  —  M')  —  (L  —  L')D.  This  gives  the  time  required  for 
the  star  to  pass  oyer  a  space  equal  to  one  revolution  of  the 
micrometer,  which  multiplied  by  15  cos  d  gives  the  value  in 
seconds  of  arc. 

We  might  compute  z  —  #„  directly  for  each  observation 
by  (475).  This  will  involve  a  little  more  labor  than  the 
method  outlined  above,  as  each  term  must  be  multiplied  by 
1  5  cos  #  ,  while  in  the  other  case  only  one  such  multiplication 
is  necessary. 

Example. 

278.  Polaris  was  observed  at  eastern  elongation,  1874,  June  18,  for  deter- 
mining the  value  of  one  revolution  of  the  micrometer  of  zenith  telescope 
Wiirdemann,  No.  20. 

Station:  Fort  Buford,  Dakota.  Observer:  Captain  J.  F.  Gregory. 

The  preliminary  computation  necessary  to  prepare  for  the  observation  is 
first  given,  viz.,  the  computation  of  the  azimuth,  zenith  distance,  and  time  of 
elongation  by  formulae  (474). 

For  this  purpose  the  right  ascension  and  declination  of  Polaris  are  taken  from 
the  Nautical  Almanac,  viz.  : 

a.  =    ih  i2m  6s.  4; 
S  =  88°  38'  3".  3. 
The  latitude  of  station  was       q>  =  47°  59'  7". 


PRACTICAL   ASTRONOMY. 
The  computation  is  as  follows: 


§  278. 


cos  d  =  8.37721 
cos  q>  =  9.82563 
sin  a  =  8.55158 

a  =  2°  2'  27 


sin  d  =  9.99988 
sin  q>  =  9.87097 
cos  z  =  9.87109 

2  =  41°  59'  50" 


<.ot      =  8.37733 
tan  cp  —     .04534 
cos  t  —  8.42267 


t  =  88°  29'    i" 

/  =    5h  53m  56s 

a  =    i    12    06 

a  —  t  —  19    18     10 

AT=  -    2 

Chronometer  time  of  elongation  =  a  —  t  —  /I  T  =   i9h  i8m  12* 

The  transit  of  Polaris  was  observed  over  the  micrometer-thread  at  every  half 
turn,  beginning  with  revolution  35  and  ending  with  5.5  —  sixty  transits  in  all. 
In  the  example  I  have  only  used  those  observed  at  the  even  revolutions,  as  this 
will  be  sufficient  for  illustrating  the  method  of  reduction. 


U 

U 

g 

. 

C 

No. 

i! 

Chronome- 

Level. 

41    §    O 

3  S*3 

fill 

0 

fsl 

Reduced 

t.  rt 

ter  Time. 

N. 

s. 

«       £ 

fsJJ 

O       " 

U 

Times. 

i 

35 

i8h  38™  40s.  o 

18.6 

19.1 

—  39m  32".  o 

+  ii8  8 

-5 

+  o".6 

i8»  38™  528.4 

2 

34 

41     38  .0 

18.5 

19.1 

36     34  .0 

9  -3 

—     .6 

+      -7 

41     48  .0 

3 

33 

44     32  .8 

18.6 

19.2 

33     39  -2 

7  -3 

—     .6 

+      -7 

44     40  .8 

4 

32 

47     27  .6 

18.7 

19.2 

30     44  .4 

5  -5 

—     -5 

+     -6 

47     33  -7 

5 

31 

50    24  .0 

19.0 

19.0 

27     48  .0 

4  -i 

.0 

50     28  .1 

6 

53     20  .6 

24     51  -4 

2  .9 

.0 

53     23  .5 

i 

9 

29 
28 
27 

56     13  -7 
18    59     io  .0 
19      2       4  .4 

19.0 
19.0 

19.1 
19.2 

19          2  .O 

16      7.6 

2  .O 

I  .3 

-     i 

i.S 

56     15  -8 
18    59     ii  .4 
19      2       5  .4 

10 

26 

5       o.o 

19.5 

19.1 

13       12  .O 

•4 

-f-     . 

—      -5 

4     59  -9 

ii 

25 

7     52  -3 

19.2 

19.2 

io     19  .7 

.2 

.0 

7     S2  -5 

12 

24 

io     49  .0 

19.6 

J9  3 

7     23.0 

+           -I 

+     1 

—      -4 

io    48  .7 

13 

23 

13     4i  -9 

4     3°  -1 

.O 

—  |— 

-      -4 

13     41  -5 

14 

22 

16    35  .0 

19.7 

19.5 

—     i     37  .0 

.0 

-L    . 

—         .2 

16     34  .8 

'5 

21 

19    29  .0 

19.6 

19-5 

+    i     17.0 

.0 

-+-    . 

—         .1 

19    28  .9 

16 

20 

22      21  .9 

20.  o 

19.4 

4       9-9 

.0 

"4~ 

—      -7 

22      21.  2 

17 
18 

'9 

.8 

25       l6  .3 

28     io  .6 

20.  2 
20.3 

19-3 
19.4 

1  31 

—       .1 

—           .2 

I  ; 

—      .1 

25       J5  -1 
28         9.3 

19 

17 

3i       3-9 

20.5 

19-5 

12      51  .9 

•4 

—        .2 

31          2  .3 

20 

21 

16 
15 

33     59-o 
36    .52  .6 

20.6 

19-5 

IS     47-0 
18    40  .6 

.8 

I   .2 

—        .2 

—      -4 

33     57-0 
36     50  .0 

22 

14 

39    46  .0 

21     34  .0 

I   .9 

—      -4 

39     42  -7 

23 

13 

42     40.0 

24     28  .0 

2  .8 

—      -4 

42     35  -8 

24 

12 

45     35  -4 

20.7 

19-5 

27     23  .4 

3  -9 

—      -5 

45     3°   o 

25 

11 

48     29  .0 

21.0 

30     17  .0 

5  -2 

-     -7 

—      .1 

48     21  .7 

26 
27 

10 

9 

Si     25  .0 
54     *9  -7 

21.0 
21  .1 

19-5 
19.4 

33     13  -o 

36       7  -7 

6.9 
9.0 

-     -5 
~     -7 

-      -9 
—      .1 

51     16  2 
54      8.6 

28 

8 

19    57     H   7 

21.0 

19  6 

39      2  .7 

«  -3 

-     -4 

—      -7 

57       *  -7 

29 

7 

20      o     13  .6 

21.0 

19.7 

42      i  .6 

14.1 

-     -3 

—      .6 

19    59     57  -9 

6 

20      3      8.6 

21  .0 

19.8 

+  44     56.6 

-  17.2 

.2 

—      -5 

20      2     49  .9 

2/8.       DETERMINATION  OF  MICROMETER    VALUE.  497 


The  first  five  columns  require  no  explanation.  The  sixth  contains  the  quanti- 
ties which  we  have  called  T.  The  "reduction  to  vertical  "  is  taken  from  the  table 
Art.  275.  The  "reduction  to  mean  state  of  level"  is  (n  —  s)  —  (n0  —  Jo), 
where  («0  —  SQ)  =  o  in  this  case.  The  "correction  for  level"  is  this  quantity 

multiplied  by  -      — 5.      The  value  of  one  division  of  the  level,   d  =  ".893. 

Therefore  this  factor  equals  1.25. 

The  elongation  being  east,  the  sign  of  the  level  reduction  is  minus. 

The  "  reduction  to  vertical"  and  "correction  for  level"  being  applied  to  the 
observed  time,  we  have  the  "  reduced  times"  of  the  last  column.  We  combine 
these  quantities  by  subtracting  No.  i  from  16,  No.  2  from  17,  ...  No.  15  from 
30,  thus  obtaining  a  series  of  values  for  the  time  required  for  the  star  to  pass 
over  a  space  equal  to  15  revolutions  of  the  screw.  The  mean  of  these  quanti- 
ties multiplied  by  —  —  =  cos  d  then  will  give  the  value  of  one  revolution 
in  seconds  of  arc. 

The  numerical  work  is  as  follows:  • 


Nos. 

Time  of  15 
Revolutions. 

V. 

TV. 

16-    i 

43m  28s.  8 

3.9 

15.21 

17-2 

43    27  .1 

2.2 

4.84 

18-    3 

43    28.5 

3-6 

12.96 

19—4 

43    28  .6 

3-7 

13.69 

20-    5 

43    28  .9 

4.0 

16.00 

21-6 

43    26.5 

1.6 

2.56 

22—7 

43    26  .9 

2.O 

4.00 

23-    8 

43    24  .4 

•      -5 

•25 

24-9 

43    24  .6 

•  3 

.09 

25  —  10 

43    21  .8 

3-i 

9.61 

26  —  II 

43    23  .7 

1.2 

1.44 

27  —  12 

43    19  -9 

5-0 

25.00 

28  —  13 

43    20  .2 

4.7 

22.09 

29  —  14 

43    23.1 

1.8  . 

3-24 

30  —  15 

43    21  .0 

3-9 

15.21 

Mean     43™  24'.  93 
=  2604".  93 


\yv\  =  146*.  1 9 


log  =  3.4157961 
cos  S  =  8.3772074 
log  one  revolution  =  1.7930035 
One  revolution         62". 0874 
Correction  for  refraction          —  .0315 
Corrected  value         62".os6 


49^  PRACTICAL   ASTRONOMY.  §  2/Q. 

The  correction  for  differential  refraction  is  computed  by  the  last  of  formula 
(481),  viz., 

r  —  r   =  [6.44676]  sec2  z(z  —  z)       6.4468 
log  (z  —  z)  =  1.7930 
sec2 z  =     .2578 
log  (r  —  r'}  =  8.4976         r  —  r'  —  ".0315 

The  probable  error  is  computed  from  the  sum  of  the  squares  of  the  residuals 
in  the  last  column  by  formula  (27),  viz., 


=  -6745  , 

111  (ni  — 

m  in  this  case  being  15.     Substituting  in  this  formula,  we  find 

r0  =  ".563. 

This  is  now  the  probable  error  of  the  determination  of  the  time  required  for  the 
star  to  pass  over  15  revolutions  of  the  screw.  The  probable  error  of  the  above 
determination  of  the  value  of  one  revolution  of  the  screw  will  be  obtained  from 

this  quantity  by  multiplying  by  the  factor  —          —  =  cos  d,  viz.,  ±  ".013. 

From  this  series  we  therefore  conclude  the  most  probable  value  of  one  revo- 
lution of  the  screw  to  be 

J?  =  62". 056  ±  ".013. 

Value  of  One  Division  of  Level. 

279.  An  example  has  been  given  (Art.  164)  of  the  determination  of  the  level 
value  by  means  of  the  level-trier.  Opposite  is  given  an  example  of  the  deter- 
mination of  the  level  value  of  the  above  instrument  by  means  of  the  microme- 
ter. See  equation  (477). 


§  280.       DETERMINATION  OF  MICROMETER    VALUE.  499 

1873,  June  15.     Observer,  L.  Boss.     Mark  cross-threads  of  transit  telescope. 


. 

Level 

Level 

<M 

"o  _• 

S    ">   O 

B"°  o 

ist  position. 

2d  position. 

rt  hr"" 

o«|. 

d 

No. 

O    rt  'w 

-  *^'o 

O    N'^ 

IL£ 

*£r 

JP 

V. 

w. 

a~& 

i  & 

N. 

S. 

N. 

S. 

u 

5s 

i 

21.036 

21.542 

13-5 

44-9 

49.1 

9-3 

35-6 

50.6 

.421 

.019 

.000361 

2 

21.538 

22.131 

8.2 

49-7 

51-8 

5-9 

43-7 

59-3 

•357 

.083 

6889 

3 

27.097 

27.650 

7-6 

49.8 

44-7 

12.4 

37-25 

55-3 

•485 

•°45 

2025 

4 
5 
6 

26.752 
19.825 
20.361 

27-387 

20  .  386 
20    889 

5.1 
6.9 
6-3 

49.0 

49-2 
43-9 
44-5 

11.4 
10.8 

45-45 
37-i 
38-2 

1? 

52.8 

.429 
.512 
382 

.Oil 

.072 
'-058 

121 

3364 

7 

20    852 

21-445 

5-3 

49-9 

48.1 

68 

42-95 

59-3 

-381 

•°59 

3481 

8 

21.438 

21.986 

9-5 

45-5 

48.2 

6-5 

38.85 

54-8 

.411 

.029 

84I 

9 

21.992 

22-555 

5-7 

48.9 

44.1 

10.3 

38.5 

56-3 

.462 

.022 

484 

10 

22.548 

23.058 

7-7 

46.6 

42.8 

35.1 

51.0 

•453 

.013 

I09 

ii 

I4-532 

13.910 

49-3 

5-1 

6.1 

48^2 

43-15 

62.2 

.441 

.001 

I 

12 

I3-903 

I3-4I3 

43-2 

10.7 

IO.2 

43-5 

32-9 

49.0 

.489 

.049 

24OI 

*3 

I3-4I5 

12.828 

48.1 

5-6 

7-6 

46.0 

4°  -45 

58.7 

•451 

.Oil 

121 

J4 

17.146 

17.670 

8.1 

45-5 

44-3 

8.9 

36.4 

52.4 

.440 

.000 

'  O 

15 

24.822 

25.310 

7.1 

45-9 

40.0 

12.9 

32-95 

48.8 

.481 

.041 

1681 

16 

25-944 

26.537 

5-2 

47-5 

46-3 

6.1 

41-25 

59-3 

.438 

.002 

4 

\yv\  =  .027127 


Mean  value  of  —  r  =  1.4396  ±  .0071. 
K 


The  above  value  of  R'  is  ".62056. 
Therefore 


d  =    ".893   ±  .004. 


If  both  the  level  and  micrometer  values  were  unknown,  the  above  series  of 
observations  of  Polaris  would  give  for  one  division  of  the  micrometer,  by  neg- 
lecting the  level  readings,  R'  =  ".6209,  which  gives  practically  the  same  value 
of  d  as  above. 

With  this  value  of  d  the  level  corrections  would  then  be  computed  and  the 
final  value  of  the  micrometer  determined,  no  second  approximation  to  the  value 
of  d  being  required. 

280.   For  the  purpose  of  illustrating  the  method  of  Art.  277  let  us  apply  it  to 

the  example  already  solved.     The  first  part  of  the  computation  will  be  precisely 

.the -same  as  before  except  the  correction  for  level.     Applying  to  the  observed 

chronometer   times   the   "reduction  to. vertical"  already  found,  we  have  the 

"  reduced  times"  of  the  following  table  : 


500 


PRACTICAL   ASTRONOMY. 


§280. 


u 
V 

Level. 

£ 

*_ 

3 

o"o 

No. 

u 

a 

O 

Reduced 
Times. 

Nos. 

1 

1 

*5 

Times. 

III 

i'. 

w. 

O 

N. 

S. 

^ 

^ 

o^ 

H  « 

i 

^^ 

•f 

0 

i 

35 

i8h38m5i8.8 

18.6 

I9.I 

16-  i 

4-   -55 

.0079 

15.0079 

2610".  i 

173s.  92 

25 

625 

2 

34 

18  41   47  -3 

18.5 

ig.I 

17—  2 

4-   -75 

.0108 

15.0108 

2608  .9 

173  -80 

13 

169 

3 

33 

44   40  .1 

18.6 

19.2 

18-  3 

+   -75 

.0108 

15.0108 

2610  .  3 

173  -9° 

23 

529 

4 

S2 

47    33-i 

18.7 

I9.2 

19-  4 

4-  .75 

.0108 

15.0108 

2610  .4 

i73  -90 

23 

529 

5 

31 

50   28  .  i 

19.0 

I9.0 

20-  5 

4-  -50 

.0072 

15.0072 

2610  .  i 

173  -92 

25 

625 

6 

3° 

53    23  .5 

21—    6 

--   -55 

.0079 

15-0079 

2607  .9 

Z73  -77 

IO 

100 

7 

29 

56    15  -7 

19.0 

I9.I 

22—    7 

--   .60 

.0086 

15.0086 

2608  .4 

173  -88 

21 

441 

8 

28 

18  59    ii  .3 

23-  8 

--   .60 

.0086 

15.0086 

2605  .9 

173  -63 

4. 

16 

9 

27 

19     2      5  .2 

19.0 

I9.2 

24-  9 

--   .70 

.OIOI 

15.0101 

2606  .3 

173  -64 

3 

9 

10 

26 

50-4 

19.5 

I9.I 

25—10 

--   -65 

.0094 

15-0094 

2603  .4 

J73  -45 

23 

529 

it 

25 

7    S2  -5 

19.2 

I9.2 

26  —  ii 

--   -75 

.0108 

15.0108 

2605  .6 

173  -58 

9 

81 

12 

24 

10   49  .1 

19.6 

27—12 

4-  -70 

.0101 

15.0101 

2601  .6 

173  -32 

35 

1225 

*3 

23 

J3   41  -9 

28—13 

f   -55 

.0079 

15.0079 

2601  .5 

173  -34 

33 

1089 

22 

16   35.0      19.7 

19-5 

29-14 

f   -55 

.0079 

15.0079 

2604  .5 

J73  -54 

169 

15 

21 

19  29  .0  I   19.6 

30—15 

+   -55 

.0079 

15.0079 

2602  .4 

T73  -40 

27 

729 

16 

20 

22     21  .9       20.0 

19.4 

X7 

19 

25   16  .2 

2O.  2 

19  3 

6865  =  \vv]. 

18 

18 

28    10  .4 

20-3 

19.4 

JQ 

17 

31     3  -5 

20-5 

19  •  5 

20 

16 

33    58  .2 

Mean  time  of  one  revolution  =  173".  666  ±.0385. 

21 

15 

36    51  -4 

20.6 

19-5 

22 

14 

39   44  -1 

23 

13 

42    37-2 

The  value  of  one  revolution  is  now  found  by 

24 

25 

12 
II 

45    31  -5 
48    23.8 

20.7 

21.0 

19.5 
19-3 

multiplying  this  time  by  15  cos  d,  viz., 

26 

IO 

51    18  .1 

21  .O 

19-5 

27 

28 

I 

54   10.7 
57     3  '4 

21.  1 
21  .O 

19.4 
19.0 

R  =  62".  0888  ±  .0138. 

29 

3° 

I 

i9  59   59  -5 

20       2     51   .4 

21.0 
21.0 

19  7 
19.8 

Refraction  =         -0315. 

Final  value  of  R  =  62".  057     ±  .014. 

If  the  chronometer  employed  has  an  appreciable  rate  the  interval  of  time 
corresponding  to  one  revolution  of  the  screw  will  require  a  correction  which 
may  be  determined  as  follows  : 

Let  8T  =  the  daily  rate  of  the  micrometer,  -j-  when  losing  ; 
/i  =  any  interval  expressed  in  terms  of  chronometer  ; 
/  =  true  value  of  interval. 

Then  /  :  /i  =  24'*  :  24*'  —  8  T  =  86400"  :  86400"  —  d  T ; 

i  dT 


/=/! 


I    — 


dT 


86400 


nearly. 


86400 

If,  for  example,  the  above  observations  had  been  made  with  a  mean  time 
chronometer,  for  d  T  we  should  have  3m  56"  =  236s.     Therefore 


/=/!+/! 


236" 
86400 


=  /i  +  /. 002735  =  173". 666  -f  .474  =  I74M40. 


*  When  the  reduction  is  made  in  this  manner  the  term  (L1  —  L)D  will  be  ±  for 
tion. 


elonga- 


§ 28  1.  FORMULAE  FOR  LATITUDE.  $01 

General  Formula  for  tJie  Latitude. 

281.  Let  m  =  the  micrometer  reading  for  the  south  star, 

expressed  in  seconds  of  arc  ; 
m*  =  the  micrometer  reading  for  the  zero-point 

of  the  micrometer  ; 
/  =  the  correction  for  level,  plus  when  the  north 

reading  is  large  ; 
r  =  the  correction  for  refraction. 

Then         z  —  z,  +  (m   -  -  m.)  +  /  +  r  for  south  star. 
Similarly^'  =  2.  +  (m'  —  m0)  —  I'  +  r'  for  north  star. 


z  -  z>  =  (m  -  „')  +  (/  +  /')  +  (r-  r'). 
Substituting  this  value  in  equation  (472), 

+  K'  -  >*')•  (478) 


It  has  been  assumed  in  the  foregoing  that  the  readings  of 
the  micrometer  increase  with  the  zenith  distance;  but,  whether 
they  increase  or  diminish,  practically  a  case  wilt  very  seldom 
occur  where  the  algebraic  sign  of  the  term  %(m  —  m'*)  will 
be  in  doubt,  as  may  be  seen  by  referring  to  the  numerical 
example. 

Equation  (478)  shows  that  the  value  of  the  latitude  is  found 
by  adding  to  the  mean  of  the  declinations  of  the  two  stars 
three  corrections:  first,  the  correction  for  micrometer; 
second,  the  correction  for  level  ;  third,  for  refraction. 


*  Any  point  may  be  assumed  arbitrarily  as  the  zero-point,  for  by  referring  to 
equations  (478)  and  (479)  it  will  be  seen  that  only  the  difference  of  micrometer 
readings  on  the  two  stars  is  required,  and  this  will  be  the  same  wherever  we 
assume  the  zero  to  be.  It  will  be  convenient  to  assume  this  point  so  far  to  one 
end  of  the  scale  that  the  readings  will  all  be  plus. 


$02  PRACTICAL   ASTRONOMY.  §  284. 

282.  The  Correction  for  Micrometer. 

Let  M  and  M'  =  the  micrometer  readings  for  the  south 

and  north  stars  respectively  ; 
R  =  the  value  of  one  revolution  of  the  screw 
expressed  in  seconds  of  arc. 

Then  #m  -  m')  =  %R(M  -  M').     ....    (479) 

If  the  micrometer  reads  towards  the  zenith  the  algebraic 
sign  will  simply  be  reversed. 

283.  The  Correction  for  Level.     If  the  mean  of  the  north 
readings  in  both  positions  of  the  instrument  is  greater  than 
the  mean  of  the  south  readings,  it  shows  that  the  vertical  axis 
produced  pierces  the  celestial  sphere  south  of  the  zenith; 
therefore  the  instrumental  zenith  distance  of  a  south  star  is 
too  small,  and  of  a  north  star  too  large. 

Let  n  and  *  —  readings  of  north  and  south  ends  of  bubble 

for  south  star  ; 
n'  and  s'  =  readings  of  north  and  south  ends  of  bubble 

for  north  star; 
y  —  the  error  of  the  level  ; 

d  =  the  value  of  one  division  of  the  level  in  sec- 
onds of  arc. 

Then  /  =  \d(n  -  s)  +  x\ 

l>  =%d(nf  -sf)  -  x', 

+  n')-(s  +  s%    .     .     (480) 


284.  Correction  for  Refraction.  The  difference  of  zenith 
distance  is  so  small  that  nothing  is  gained  by  applying  to 
the  correction  for  refraction  the  terms  depending  on  the  ba- 
rometer and  thermometer. 


§284.  FORMULA  FOR  LATITUDE.  503 

Bessel's  formula  for  mean  refraction  is 

r  =  a  tan  z .     (a) 

a  for  present  purposes  is  considered  constant  and  equal  to 

57"7. 

The  correction  r  —  r'  being  very  small,  we  may  use  a  dif- 
ferential formula,  viz., 

df        ,  X  »v 


j 
and  from  (a),  -^  —  $?".7  sec*z. 

If  z  —  z'  is  given  in  minutes  we  may  write  (b)  as  follows: 

r  —  r'  =  57". 7 sec2  z.s\n  if .(z  —  #'), 
or  r  —  r'  —  [8.22491]  sec2  z  .(z  —  z'\ 

If  (z  —  z')  is  expressed  in  seconds, 

(r  —  r')  =  [6.44676}  sec2  z.(z  —  z'\ 

As  usual  the  numerical  quantities  in  brackets  are  logarithms. 
The  computation  by  either  of  these  formulas  is  quite 
simple,  but  as  this  correction  must  be  applied  to  every  pair 
of  stars  observed  the  following  table  has  been  added,  being 
the  same  as  that  given  by  Schott,  of  the  U.  S.  Coast  Survey. 
The  vertical  argument  is  one  half  the  difference  of  zenith 
distance,  for  which  we  may  use  %(m  —  m').  The  horizontal 
argument  is  the  zenith  distance,  the  table  being  extended  to 
35°.  In  the  exceptional  cases  where  stars  are  observed  at 
greater  zenith  distances  the  correction  must  be  computed  by 
the  formula  (481).  The  algebraic  sign  will  always  be  the 
same  as  that  of  the  micrometer  correction. 


504  PRACTICAL  ASTRONOMY. 

TABLE   B— DIFFERENTIAL   REFRACTION. 


§285, 


Half  Difference  in 
Zenith  Distance. 

Zenith  Distance. 

0° 

IO° 

20° 

25° 

30° 

35° 

O 

oo 

.00 

.00 

.OO 

.OO 

.00 

0-5 

.01 

.OI 

.OI 

.01 

.01 

.01 

I 

.02 

.02 

.02 

.02 

.02 

.02 

1-5 

•03 

•03 

•03 

.03 

•03 

•03 

2 

•03 

•03 

.04 

.04 

.04 

•05 

2-5 

.04 

.04 

•05 

•05 

•05 

.06 

3 

•05 

•05 

.06 

,06 

.07 

.08 

3-5 

.06 

.06 

.07 

.07 

.08 

.09 

4 

.07 

.07 

.08 

.08 

.09 

.IO 

45 

.08 

.08 

.09 

.09 

.10 

.11 

5 

.08 

.09 

.  IO 

.  IO 

.  II 

.13 

5-5 

.09 

.10 

.  IO 

.  II 

.12 

•14 

6 

.10 

.10 

.11 

.12 

•13 

•15 

6.5 

.  II 

.11 

.  12 

•13 

.14 

.16 

7 

.12 

.12 

•13 

.14 

•15 

.18 

7  5 

•13 

•13 

.14 

•15 

.16 

.19 

8 

•13 

.14 

•15 

.16 

.18 

8-5 

.14 

•15 

.16 

•17 

.19 

.22 

9 

•15 

.16 

•17 

18 

.20 

•23 

9-5 

.16 

•17 

.18 

.20 

.21 

.24 

10 

•17 

.18 

.19 

.21 

•23 

.26 

10.5 

.18 

.19 

.20 

.22 

.24 

•27 

IT 

.18 

.19 

.21 

•23 

•25 

.28 

ii.  5 

.19 

.20 

.22 

.24 

.26 

•30 

12 

.20 

.21 

•23 

•25 

.27 

•31 

12.5 

.21 

.21 

.24 

.26 

.28 

•32 

285.  Reduction  to  the  Meridian.  If  the  observation  has  been 
missed  at  the  instant  of  the  star's  meridian  passage,  it  may 
be  observed  off  the  meridian  in  either  of  two  ways : 

First.  The  instrument  may  be  revolved  in  azimuth  so  as  to 
bisect  the  star  in  the  middle  of  the  field;  or 

Second.  The  instrument  may  be  allowed  to  remain  in  the 
meridian,  and  the  star  may  be  bisected  off  the  line  of  colli- 
mation  before  it  passes  out  of  the  field. 

In  the  first  case  the  correction  to  the  zenith  distance  will 
be  precisely  the  same  as  that  already  derived  for  reducing 


§286. 


REDUCTION   TO  MERIDIAN. 


505 


circummeridian  altitudes,  viz. — see  equations   (XIII),    Art. 
149— 

cos  tp  cos  d  2  sin2  \t 


sm  z 


sm  i' 


where  t  is  the  hour-angle  of  the  star  at  the  instant  of  observa- 
tion. 

The  quantity  given  by  this  formula  is  to  be  subtracted  from 
the  zenith  distance  at  the  instant  of  observation ;  therefore 
by  referring  to  (472)  we  see  that  the  correction  to  the  latitude 
will  be 

i  cos  cp  cos  $  2  sin2  \t  . '     . 

^=±a— ^El--ilE-F--  '    '    '    (482) 


will  be   plus  for  a   north  and  minus  for  a  south  star. 

— = ^r  is  taken  from  table  VIII  A  at  the  end  of  this  volume. 

sm  i 

286.  When  the  star  is  observed  off  the  line  of  collimation,  the 
instrument  remaining  in  the  meridian.  In  the 
figure,  PK  is  the  meridian,  PS  the  hour- 
circle  passing  through  the  star.  If  the  star 
is  observed  on  the  meridian,  SAT  will  be  the 
position  of  the  micrometer-thread.  If  ob- 
served off  the  meridian  at  S',  this  thread  will 
have  the  position  S'K'. 

Let  KK1  =  x. 
Then  PK  =  90°  -  (6  +  x\ 

and,  by  Napier's  second  rule, 


cos  t  =  tan  $  cot  (tf  +  x]. 


506  PRACTICAL  ASTRONOMY.  §  286. 

This  may  be  placed  in  the  form 

tan  #  4-  tan  ;r 


,  . 

tan  d  =  (i  -  2  sin8 


—        ~-  ---  . 
I  —  tan  a  tan  .r 


Clearing    of     fractions     and     neglecting     the    small    term 
tan  x.2  sin2^/,  we  readily  find 

tan  x  —  sin  d  cos  d  2  sin2^/, 
or,  with  sufficient  accuracy, 

.  2  sin2  \t 
x  ==  ism  2#    sin  t/,  ......  (483), 

As  the  apparent  zenith  distance  is  diminished  for  a  south 
star  and  increased  for  a  north  star  when  observed  in  this  man- 
ner, the  correction  to  the  latitude  will  always  be  plus  and 
will  be  equal  to  %x.  That  is, 

A9  =  i  sin  26  2*™*f  ......    (483) 

This  method  of  proceeding  will  generally  be  preferred 
when  the  observation  on  the  meridian  is  lost,  as  when  the 
other  method  is  used  the  stop  must  be  undamped,  and  where 
other  stars  follow  in  quick  succession  a  pair  may  be  lost  in 
consequence.  If  the  star  cannot  be  observed  before  it  gets 
beyond  the  field  of  view,  the  observer  will  generally  prefer 
to  let  it  go  altogether. 

The  computation  of  Ay  by  the  above  formula  is  very 
simple,  but  a  table  is  added  from  which  the  value  of  x  =  2Acp 
may  be  taken  at  once.  The  horizontal  argument  is  the  hour- 
angle  of  the  star,  and  the  vertical  argument  the  declination. 


§  288.  COMBINATION  OF  RESULTS. 

TABLE  C— REDUCTION  TO  MERIDIAN. 


507 


ioy. 

15*- 

2ay. 

25J. 

3ay. 

35*- 

40*. 

45-r- 

5os. 

55*- 

6ar. 

d 

II 

II 

II 

II 

II 

It 

ti 

n 

II 

It 

// 

d 

5° 

.00 

.OI 

.02 

•03 

.04 

.06 

.08 

.10 

.12 

.14 

•17 

85° 

10 

.OI 

.02 

.04 

.06 

.08 

.11 

•15 

.19 

•23 

.28 

•34 

80 

15 

.01 

•03 

•05 

.09 

.12 

•17 

.22 

.28 

•34 

.41 

.49 

75 

20 

.02 

.04 

.07 

.  II 

.16 

.22 

.28 

.36 

.44 

•  53 

.63  !  70 

25 

.02 

•05 

.08 

•13 

.19 

.26 

•34 

.42 

•  52 

•63 

•75 

65 

30 

.02 

•05 

.09 

•15 

.21 

.29 

.38 

.48 

•59 

•71 

.85 

60 

35 

•03 

.06 

.10 

.16 

•23 

•31 

.41 

•53 

.64 

•  77 

.92 

55 

40 

•03 

.06 

.11 

•17 

.24 

•33 

•43 

•54 

.67 

.81 

•97 

50 

45 

•03 

.06 

.11 

•17 

•25 

•33 

•44 

•  55' 

.68 

.82 

.98 

45  - 

287.  Formula  for  Computation  of  Latitude  from  Observations 
with  the  Zenith  Telescope. 


•—  r')  =  [8.22491]  sec2  z  \(z  —  z 

c 

Reduction  to  Meridian. 

1  cos  (p  cos  8  2  sina  \ 

A<p  =  ±  -  -          : ^ 

2  smz        sin  i 

i     .        .  2  sin2  \t 

f  dcp  =  4-  -  sin  2(^  —. 77-. 

1    4  sin  i" 


N. 


star; 


(XXIII) 


Combination  of  the  Individual  Values  of  the  Latitude. 

288.  For  many  purposes  a  sufficient  degree  of  accuracy 
will  be  given  by  simply  taking  the  arithmetical  mean  of  the 
individual  values,  giving  all  equal  weight. 


See  table,  p.  504. 


f  See  table  above. 


508  PRACTICAL   ASTRONOMY.  §  288. 

When  a  more  rigorous  procedure  is  demanded  we  must 
consider  the  weights  of  the  separate  values.  This  weight 
depends  on  the  probable  errors  of  the  declinations  of  the 
stars  observed,  and  on  the  probable  error  of  observation. 

Let  /  =  the  number  of  separate  pairs  employed 

in  determining  a  latitude ; 
nv  #a»  #3»  •  •  •  np  =  the  number  of  observations  on  each 

pair  respectively  ; 
n— #!+»„+  •  •  •  ~\~np  —  the  whole  number  of  observations ; 

e  =  the  probable  error  of  a  single  observa- 
tion. 

Then,  from  (35), 


(n,  -  \)ee  =  ( 

(n,  -i)ee  =  (.6745)^1 J; 


(np-i)ee  =  (. 
The  sum  of  these  equations  gives 
(n  -£)ee=  (. 


therefore  e  =  .6.745\/  ;rir7 (4»4) 


[z/,^j]  is  the  sum  of  the  squares  of  the  residuals  formed  by 
taking  the  differences  between  the  mean  of  the  observations 
on  the  first  pair  and  each  individual  value;  and  similarly  for 


§289.  COMBINATION  OF  RESULTS.  509 

The  determination  of  the  probable  errors  of  the  decima- 
tions is  a  much  more  complicated  problem.  For  a  discussion 
of  this  subject  the  reader  will  refer  to  Articles  346  and  347. 

In  order  to  obtain  the  expression  for  the  weight  of  the 
value  of  cp  derived  from  a  single  pair, 

Let  ffi,  e#  =  the  probable  errors  of  the  declinations; 


Then  if  nl  is  the  number  of  observations  on  this  pair  the 

/? 

probable  error  of  the  mean  will  t>e  A  /  -, 


and 


^    


E+  being  the  probable  error  of  the  resulting  latitude. 

The  relative  weights  are  proportional  to  the  reciprocals  of 
the  squares  of  the  probable  errors;  or,  since  the  unit  of  weight 
is  arbitrary,  we  may  write 


(485) 


Value  of  Micrometer  from  the  Latitude  Observations. 

289.  If  no  special  observations  have  been  made  for  deter- 
mining the  value  of  the  micrometer-screw,  it  maybe  derived 
from  the  latitude  observations  themselves. 

*  Equation  (29). 


510  PRACTICAL   ASTRONOMY.  §  290. 

Let  R  =  an  assumed  value  of  one  revolution  as  near 

the  true  value  as  possible; 
AR  =  the  correction  required. 
Then  R-\-  AR  =  the  true  value  of  one  revolution; 

<pr  =  the   latitude  computed   with   the  assumed 

value  of  R  from  all  of  the  observations; 
<pf  -f-  Ay  =  true  value  of  the  latitude. 


Then  from  (478), 


Let  n  =  the  sum  of  the  known  quantities  of  this  equation; 
that  is,  n  =  <pf-%(d+d')-±R(M-M')-%(l+r)-l(r-r'). 
Then  A9  -  \(M  -  M')AR  =  n  .....     (486) 


Each  pair  of  stars  observed  will  give  an  equation  of  this 
form  for  determining  A<p  and  AR. 

This  process  is  sometimes  employed  when  there  is  reason 
to  suspect  that  the  adopted  value  of  R  is  erroneous;  but  if  the 
value  has  been  carefully  determined  by  the  transits  of  cir- 
cumpolar  stars  the  result  will  generally  be  accepted  as  ab- 
solute. 

290.  The  example  which  follows  is  taken  from  the  report 
of  the  U.  S.  Northern  Boundary  Survey.  The  station  is  47 
miles  west  of  Pembina,  the  approximate  position  being 

Latitude  49°  oo',  Longitude  ih  24m  52"  west  of  Washington. 


§  2QO.       EXAMPLE   OF  LATITUDE  DETERMINATION.  511 

it 

Twenty-nine  pairs  of  stars  were  observed  from  two  to  five 
times  each,  in  all  81  observations. 

The  form  in  which  the  example  is  given  will  be  found  a 
convenient  one  for  the  record  and  preliminary  reduction. 
For  this  purpose  a  book  will  be  required  with  a  page  of  about 
7  inches  in  width.  It  will  be  ruled  or  printed  in  blank  form 
as  shown. 


Example, 

Astronomical  Station  No.  4. — West  side  of  Pembina  Mountain. 

Observer,  Lewis  Boss. 
Zenith  Telescope,  Wurdemann  No.  20.     Chronometer,  Negus  Sidereal  No.  1513. 


Date. 

Star. 
B.  A.  C. 

N. 
or 
S. 

Micro- 
meter 
Reading. 

M  -  M'. 

Level. 

N  -  S. 

Meridian 
Distance 

8. 

N. 

S. 

1873. 

June  27. 

4937 

N. 

.28.191 

30-9 

25-7 

50°    9'    o".77 

4974 

S. 

10.209 

-  17.982 

39-2 

19.0 

+  25-4 

48      9     4   .70 

5026 

S. 

18.927 

31-0 

27.2 

38    44  32   .88 

5097 

N. 

28.265 

-    9-338 

29.2 

32.0 

+     I.O 

59    24  48   .13 

527i 

S. 

21.628 

27.0 

28.7 

42    48   31   .36 

53i3 

N. 

17.220 

+    4-408 

28.3 

27.1 

-    o-5 

55      6   37   .48 

5415 

N. 

27.762 

29-5 

25.6 

58    16   13   .36 

5460 

S. 

ii  .010 

—  16.752 

27.0 

28.3 

+     2.6 

40     o  50   .34 

5502 

N. 

9.401 

32.4 

22.3 

55    29  43   .30 

5523 

S. 

29.009 

+  19.608 

21.5 

34-o 

-  2.4 

42      9   45   -25 

5853 

N. 

25.158 

31.0 

26  2 

49    49   41    .09 

59" 

S. 

13-555 

—  ii.  603 

24-3 

33-2 

—   4.1 

48    22      i    .81 

6047 
0073 

N. 
S. 

26.368 
9.814 

-  16.554 

34-o 

22.0 

23-4 
35-4 

-    2.8 

72    12   35    .57 
26      4    15    .06 

6114 

N. 

12  .071 

28.5 

28.3 

59' 

76    58    38    .19 

i 

6i57 

S. 

25.001 

+  12.930 

27.1 

30-4 

-    3-i 

20    47    41    .84 

6268 

S. 

14.417 

26.9 

3J-3 

39    a6    16   .95 

6289 

N. 

24.251 

"    9-834 

30-5 

27-4 

-    i-3 

16 

58    43   35    -54 

June  29. 

5271 
53i3 

S. 

N. 

20.694 
16.306 

+    4-338 

25-5 
32.1 

3°-9 
24.8 

+    1.9 

42    48   32    .48 
55      6   37   .90 

5415 

N. 

27-4I3 

3°-5 

26.3 

58    16   13   .78 

5460 

S. 

10.648 

-  16.765 

27.8 

29.6 

+    2.4 

40      o   50    .83 

55°2 

N. 

9.152 

29.6 

28.0 

55    29   43    .76 

5523 

S. 

28.712 

-f  19.560 

27.8 

30.2 

-    0.8 

42      9   45    .68 

512 


PRACTICAL   ASTRONOMY. 


§290. 


(S  +  *')  and 
K*  +  *')• 

CORRECTIONS. 

Latitude. 

Remarks. 

Micrometer. 

Level. 

Refrac- 
tion. 

Merid- 
ian. 

98°  18'    s".47 

49      9     2   .74 

—    9'  i7"-94 

+  5-69 

-  .16 

48°  59'  5o".33 

98      9  21    .01 

49      4   40   .50 

—    4   49   -73 

+      .22 

-  .08 

50  .91 

97    55     8    .84 
48    57    34    -42 

+    2    16   .77 

—      .11 

+  .04 

51     .12 

98    17      3    .70 

49      8    31    .85 

—    8   39   .77 

+      .58 

-  -15 

52    -51 

97    39    28    -55 

48    49   44    .28 

+  10     8   .40 

-     -53 

+  .18 

52    -33 

98    ii    42    .90 

49      5    5'    -45 

—      6       0     .02 

-     .91 

—   .10 

5°   -42 

98    16  50  .63 

49      8   25   .32 

-    8   33    .62 

-     .62 

-  .16 

50   .92 

97    46   20    .03 

48    53    10    .02 

+    6  41    .19 

-     .69 

+  .14 

+  .21 

50  .87 

Reduction  to 
meridian   taken 

98      9    52    .49 

from    table    C, 

49      4    56   .25 

-     5      5-^3 

-     .29 

—  .09 

+  .03 

50   -77 

Art.  286. 

97    55    10   -38 
48    57   35    -19 

-f-    2   16  .15 

+    -42 

+  .04 

51    .80 

98    17     4   .61 

* 

49      8   32   .31 

-    8   40   .17 

+    -53 

-  -15 

52   .52 

97    39   29   .44 

48    49    44    .72 

+  10     6    .90 

-     .18 

+  .18 

51    .62 

The  above  probably  requires  no  further  explanation  than  a  reference  to 
formula  (XXIII),  Art.  287. 

The  values  of  the  micrometer-screw  and  level  which  we  have  employed  are 
those  derived  in  Articles  278  and  279,  viz., 

^  =  62".o56; 
d  =    o   .893. 


This  will  be  sufficient  for  illustrating  the  method  of  reduction.  In  order  to 
illustrate  the  combination  of  the  individual  values  to  determine  the  most  prob- 
able value,  the  weights  and  probable  errors,  the  results  of  the  entire  81  obser- 
vations will  be  employed.  They  are  as  follows: 


290. 


LA  TITUDE  DE  TERM  IN  A  TION. 


513 


Star. 
B.  A.  C. 

Date. 
June. 

Lati- 
tude. 

48°  59' 

Mean. 

48°  59' 

V. 

vv. 

Star. 
B.A.  C. 

Date. 
June. 

Lati- 
tude. 
48°  59' 

Mean. 

48°  59' 

V. 

vv. 

4937 
4974 

26 
77 

5o".87 

5°  -33 

5o".6o 

27 

27 

729 

729 

6047 
6073 

25 
26 

5i"-33 
5i  -53 

2 
22 

4 

484 

5026 
5097 

26 

27 

Si  -59 
50  .91 

51  -25 

34 
34 

1156 

1156 

27 
3° 

50  .92 
51  .46 

5i".3i 

39 
IS 

1521 
225 

5271 
53i3 

25 
26 

27 
29 

51  .42 
51  .10 

51  -12 
SI  .80 

5i  -36 

06 
26 
24 
44 

36 
676 
576 
1936 

6157 

25 
26 
27 
29 
3° 

51  .16 
52  -44 
50  .87 
51  .64 
52  .31 

51  .68 

S2 
76 
81 
4 
6^ 

2704 
5776 
6561 
16 
3969 

54*5 
5460 

25 
26 
27 

29 

Si  -58 
Si  -63 
52  .51 
52  -52 

52  .06 

48 
43 
45 
46 

2304 
1849 
2025 
2116 

6268 
6289 

25 
26 

27 
29 

50  .36 

50  -53 
5°-  -77 
50  .72 

46 
29 
5 

10 

2116 

841 
25 

IOO 

5502 
5523 

25 

52  -56 

47 

2209 

3° 

5i  -71 

50  .82 

89 

7921 

26 
27 
29 

51  .84 
52  -33 
51  .62 

52  .09 

25 
24 
47 

625 
576 
2209 

6318 
6365 

26 
3° 

5i  -57 
51  .24 

5i  -40 

17 

16 

289 
256 

5545 
5624 

25 
26 
29 

50  -95 
51  .68 
51  .18 

Si  -27 

32 
41 
9 

1024 
1681 
81 

6476 

25 
26 
29 
3° 

52  .98 
51  .89 
51  .86 
51  -85 

52  .14 

84 
25 
28 
29 

7056 
625 
784 
841 

5658 

26 

29 

50  .78 
51  .66 

51  -22 

44 
44 

1936 
1936 

6553 
6586 

25 
26 

52  .01 
51  -51 

26 
24 

676 

57& 

5693 
5823 

25 

52  -44 

46 

2116 

• 

30 

5i  -74 

5i  -75 

i 

i 

26 
29 

52  -44 
5i  -07 

51  .98 

46 
91 

2116 
8281 

6624 
6681 

25 

26 

Si  -13 
51  .28 

25 

10 

625 

IOO 

5853 
5911 

25 

51  .11 

24 

576 

3° 

Si  -73 

51  -38 

35 

1225 

26 
27 
29 

51  .18 
50  .42 
So  -77 

50  .87 

3i 
45 

10 

961 
2025 

IOO 

6728 
6748 

25 
26 

Si  -56 
51  -75 

51  .66 

10 
9 

IOO 

81 

PRA  CTICAL   A  STKONOM  Y. 


Star. 
B.A.  C. 

Date. 
June. 

Lati- 
tude. 

48°  59' 

Mean. 
48°  59' 

i 

„, 

Star. 
B.A.C. 

Date. 
June. 

Lati- 
tude. 
48°  59' 

Mean. 

48°  59' 

V. 

- 

6780 
6817 

25 

5i".96 

43 

1849 

7377 
7398 

26 

i" 

I 

i 

26 

51  .19 

34 

1156 

3° 

51  -99 

Si".99 

0 

o 

3° 

51  -44 

5i"-53 

9 

81 

7416 

26 

5i  -56 

9 

81 

6937 
6970 

26 

51  .50 

29 

841 

3° 

5i  -39 

5i  -47 

8 

64 

3° 

50  -93 

51  .21 

28 

784 

7480 
I  7489 

26 

53  -02 

3 

9 

1 

7024 
7°73 

26 

52  .23 

61 

3721  i 

| 

3° 

52  -97 

52  .99 

2 

4 

30 

51  .02 

51  .62 

60 

3600 

7505 

26 

51  -57 

10 

100 

7100 
7166 

26 

51  -31 

13 

169 

30 

51  -77 

5i  -67 

IO 

IOO 

30 

5i  -57 

51  -44 

13 

769 

7627 
7686 

26 

52  -55 

76 

5776 

7215 
7277 

26 

53  -27 

99 

9801 

30 

51  -03 

5i  -79 

76 

5776 

3° 

51  -29 

52  .28 

99 

9801 

7755 
7765 

26 

52  .19 

9 

81 

7320 

26 

53  -18 

92 

8464 

3« 

52  .01 

52  .10 

9 

81 

30 

51  -33 

52  .26 

93 

8649 

M 

=  i5 

.0396 

If  we  take  the  arithmetical  mean  of  the  81  determinations,  giving  equal 
weights  to  all,  we  find  as  the  result 

cp  =  48°  59'  51". 60  ±  .048. 

291.  If  we  desire  the  highest  degree  of  precision,  we  must  combine  the  val- 
ues obtained  from  the  individual  pairs  of  stars  according  to  their  respective 
weights.  The  probable  error  of  observation  is  determined  from  the  quantity 
[vv\  above  by  means  of  formula  (484),  viz. , 


e  —  .6745' 
In  this  case      n  =  81,        /  =  29; 


n-p 
therefore         e  =  ".363. 


We  shall  assume  e  =  o".4  in  computing  the  weights  by  formula  (486). 

This  computation  immediately  follows.  The  values  of  £g  are  those  given  by 
Boss  in  his  Catalogue  of  500  Stars.  In  case  of  a  few  stars  where  Boss  assigns 
no  value  to  the  probable  error,  it  has  been  assumed  to  be  o".75. 

Referring  to  formula  (485),  the  following  computation  will  be  clearly  under- 
stood 


291 


LATITUDE  DETERMINATION. 


515 


No.  of  ; 

Bsrc.0b-:  £« 

vations. 

««» 

l£a_ 

n 

/ 

<J> 

48°  59' 

/** 

V 

pvv 

i 

4937 
4984 

2 

•25 
•23 

.0625 
529 

.3200 

2.30 

50".  60 

1-38 

-  .96 

2.1197 

2 

5026 

5097 

2 

•27 

.09 

729 
81 

.3200 

2-49 

5i  -25 

3-" 

-  -3i 

•2393 

3 

5271 
5313 

4 

.26 
.22 

676 
484 

.1600 

3-62 

51  -36 

4.92 

—   .20 

.1448 

4 

54!5 
5460 

4 

•35 
•49 

1225 
2401 

.1600 

1.91 

52  .06 

3-96 

+   -50 

•4775 

5 

55°2 
5523 

4 

•  25 
.27 

625 
729 

.1600 

3-39 

S2  .09 

7.09 

+  -S3 

•9523 

6 

5545 
5624 

3 

•13 
•3° 

169 
900 

•2133 

3-12 

5i  -27 

3.96 

-  .29 

.2624 

7 

5644 
5658 

2 

•29 
.29 

841 
841 

.3200 

2.05 

51  -22 

2.50 

-  -34 

.2370 

8 

5693 
5823 

3 

.19 
.  ii 

361 

121 

•2133 

3-82 

5I  .98 

7.56 

+  -42 

.6738 

9 

5853 
59" 

4 

:S 

900 
324 

.1600 

3-54 

50  .87 

3  08 

-   69 

1.6854 

10 

6047 
6073 

4 

.11 
.14 

121 
I96 

.1600 

5-22 

51  -31 

6.84 

-  -25 

•  3263 

ii 

6114 
6i57 

5 

.18 
•  25 

3^4 
625 

.1280 

4-49 

51  .68 

7-54 

+   .12 

.0647 

12 

6268 
6289 

5 

.22 
•23 

484 
529 

.1280 

4-36 

50  .82 

3-58 

-   -74 

2.3875 

13 

6318 
6365 

2 

.21 
•25 

441 
625 

.3200 

2-34 

51  .40 

3-28 

-   .16 

•0599 

14 

642  1 
6476 

4 

•3° 

•56 

900 

3136 

.itoo 

1.77 

52  -14 

3-79 

+  .58 

•5954 

15 

16 

6553 
6586 
6624. 
6681 

3 
3 

•23 
.19 
•75 
.40 

529 

5625 
1600 

•2133 
•2133 

3-3i 
i  07 

Si  -75 
Si  -38 

5-79 
1.48 

+  -19 

-   .18 

•"95 
•°347 

17 

6728 
6748 

2 

•75 
•35 

5625 
1225 

.3200 

•99 

51  -66 

1.64 

+   .10 

.0099 

18 

6780 
6817 

3 

.28 
.41 

784 
1681 

•2133 

2.17 

5i  -53 

3-32 

-  -«3 

.0020 

19 

6930 
6970 

2 

.22 
.26 

484 

676 

.3200 

2.29 

51  -21 

2-77 

-  -35 

.2805 

20 

7024 
7073 

2 

•37 
.19 

1369 
361 

.3200 

2.03 

51  .62 

3  29 

+  .06 

.0073 

21 

7100 
7166 

2 

•75 
•75 

5625 
5625 

.3200 

.69 

51  -44 

•99 

—   .  12 

.0099 

22 

7215 
7377 

2 

•30 

.12 

000 

144 

.3200 

2.36 

52  .28 

5-38 

+.   -72 

1.2234* 

23 

7320 

2 

.24 

•75 

0576 
5625 

.3200 

i.  06 

52  .26 

2.40 

+   -70 

•  5194 

24 

7377 
7398 

2 

•29 
•*3 

841 
169 

.3200 

2.38 

5i  -99 

4-74 

+  -43 

.4401 

25 

7416 
7453 

2 

.07 
•25 

49 
625 

.3200 

2.58 

5i  -47 

3-79 

-  .09 

.0209 

26 

7480 
7489 

2 

.19 
•75 

£' 

5625 

.3200 

1.09 

52  -99 

3-26 

+  1-43 

2.2289 

27 

75°5 
7605 

2 

.14 
•3° 

i96 
900 

.3200 

2-33 

5i  -67 

3.89 

+  .« 

.0282 

28 

7627 
7686 

2 

.08 

.16 

64 

256 

.3200 

2.84 

51  -79 

5-08 

+  -23 

.1502 

29 

7755 
7765 

2 

•25 
.20 

625 
400 

.3200 

2-37 

52  .10 

4.98 

+  -54 

.  691  i  . 

[/<f>]  =  115.39 


=  15-9920 


*  In  this  column  only  the  last  three  figures  of  /<£  are  given. 


5l6  PRACTICAL   ASTRONOMY.  §292. 

The  probable  error  of  q>  is  r0  =  .6745 1/  .-  -.        _ — - (35) 

Substituting  in  this  formula,  r0  =  .059. 

The  smaller  value  of  the  probable  error  when  the  mean  of  the  81  determina- 
tions is  formed  directly  is  fallacious,  since  it  rests  on  the  assumption  that  these 
81  values  are  independently  determined.  If  each  value  were  derived  from  a 
separate  pair  of  stars  this  would  be  correct,  but  since  the  81  values  depend  on 
only  29  separate  pairs  the  error  of  the  assumption  is  obvious. 

It  might  be  a  question  whether  No.  26  should  not  be  rejected,  this  value  dif- 
fering from  the  mean  by  a  quantity  so  much  larger  than  any  of  the  others.  There 
appears  to  be  no  reason  for  its  rejection  aside  from  this  rather  large  discrepancy. 
If  we  reject  it  we  find  from  the  remaining  28  pairs 

V  =  48°  59'  51". 54  ±  .056. 

292.  For  an  illustration  of  the  method  of  Art.  289,  let  us  form  the  equations 
for  determining  the  correction  to  the  adopted  value  of  R  and  to  the  above 
value  of  (p.  We  shall  have  29  equations  of  the  form  (486);  the  above  values  of 
•v  will  be  the  absolute  terms.  If  we  refer  to  the  observations  given  in  Art.  290, 
we  have  for  the  first  pair  %(M  —  M')  =  —  8.99.  We  have  from  this  pair  the 
equation 

A(p  -j-  8.99  AR  —  n. 

This  star  was  observed  on  two  nights,  so  taking  the  mean  of  the  values  of 
\(M  —  M'}  and  multiplying  the  resulting  equation  through  by  the  square 
root  of  the  weight  determined  for  this  star,  we  have  the  following  equation: 

T..$2A(p-\-  13.  tfAR  •=   —   1.46. 

Proceeding  in  a  similar  manner,  we  derive  the  following  29  equations  of  condi- 
tion for  determining  Acp  and  AR,  for  which  we  shall  write  x  and  7: 

.52*+  13.577  =  -  1.46; 
.58*+  7-367  =  -  -49; 
.90.*  —  4.227  =  —  .38; 
.38*+  11.567  =  -f  .69; 

.84*  —  18.037  —  -f  -98; 

.77*  -  18.537  =  -  -51; 

.43*  +  4.457  =  —  -49; 

.95*  -  11.977  =  +  .82; 

.88-r  -|-  10.887  =  ~  I-3°; 


293- 


LATITUDE  AND  MICROMETER. 


517 


2.28*  + 

18.867  = 

—    -57J 

2.12*  — 

13.727  = 

+    -25; 

2.09*  + 

10.307  — 

—  1-55 

•  53*  - 

12.517  = 

—    .24; 

•33*  - 

.207  = 

+    -77; 

.82*  + 

3.697  = 

+    -35; 

.03*- 

2.997  = 

—    -19; 

.oo*  + 

2.887  = 

+    .10; 

.47*  - 

•357  = 

-    .04; 

.51*  + 

7.077  = 

-    -53; 

.42*  - 

4.697  = 

+    -09; 

.83*  + 

7.637  = 

—    .10; 

.54*  - 

8.817  = 

+  1.  11; 

.03*- 

2.817  = 

+    -72; 

•  54*  + 

14.607  = 

+    -66; 

.61*  + 

7.787  = 

—    .14; 

.04*  + 

1.217  ~ 

+  I.49J 

•  53*  + 

3  017  = 

+    -17; 

.69*  — 

4-777  = 

+    -39; 

•54*  - 

5.707  = 

+    .83- 

Proceeding  in  the  usual  manner,  we  derive  from  these  the  two  normal  equations 


From  these, 


73.98*+    17.657  =  —  004; 
17.65*  +  2732.357  =  -  85.80. 

*  =  +  .007  ±  .054; 

7  =  —  .031  ±  .009. 


The  most  probable  values  of  the  latitude  and  micrometer-screw  as  indicated 
by  this  series  of  observations  are  therefore 

(f>  —  48°  59'  51". 567  ±  .054; 
R  =  62". 025  ±  .009. 

In  order  to  have  the  value  of  R  determined  in  this  way  of  any  value  in  com- 
parison with  that  determined  by  transits  of  circumpolar  stars,  the  declinations 
of  the  stars  employed  must  be  well  determined. 

293.  There  are  various  ways  in  which  the  observation  of 
stars  in  pairs  at  equal  or  nearly  equal  altitudes  by  means  of 
the  zenith  telescope  may  be  employed  for  the  determination 


$18  PRACTICAL   ASTRONOMY.  §  293- 

of  latitude  and  time.  As  may  be  seen,  the  instrument  is 
adapted  to  the  solution  of  any  problem  of  Spherical  Astron- 
omy which  depends  upon  the  observation  of  two  or  more 
bodies  at  the  same  altitude.  The  most  favorable  condition 
for  latitude  determination  is  when  the  two  stars  are  on  the 
meridian,  one  north,  the  other  south,  while  time  is  best  de- 
termined by  observing  two  stars  on  the  prime  vertical,  one 
east,  the  other  west. 

On  account  of  the  facility  with  which  the  latitude  is  deter- 
mined in  the  manner  already  explained,  and  the  ease  with 
which  the  instrument  may  be  converted  into  a  transit  when 
it  is  necessary  to  employ  it  for  determining  the  approximate 
time,  other  solutions  of  the  problem  depending  on  observa- 
tions out  of  the  meridian  have  never  met  with  much  favor. 

Some  of  these  methods  are  interesting  from  a  theoretical 
point  of  view,  but  for  the  reasons  stated  the  subject  will  not 
be  developed  further  in  this  connection. 


CHAPTER  IX. 

DETERMINATION    OF   A'ZIMUTH. 

294.  The  Azimuth  of  a  point  on  the  earth's  surface  is  the 
angle  between  the  plane  of  the  meridian  and  the  vertical 
plane  which  passes  through  this  point  and  the  eye  of  the 
observer. 

Since  the  vertical  plane  is  determined  by  the  direction  ot 
the  plumb-line,  and  this  4ine  may  deviate  from  th«  true 
normal  to  the  earth's  surface,  a  corresponding  deviation  in 
the  azimuth  must  exist.  We  must  therefore  distinguish  be- 
tween the  Astronomical  Azimuth  and  the  Geodetic  Azimuth. 

The  Astronomical  Azimuth  of  a  point  is  the  angle  between 
two  planes  drawn  through  the  plumb-line  at  the  point  of 
observation,  the  first  plane  parallel  to  the  earth's  axis,  and 
the  second  passing  through  the  point. 

The  Geodetic  Azimuth  is  the  angle  between  two  planes 
drawn  through  the  normal  to  the.  earth's  surface  at  the  point 
of  observation,  the  first  plane  passing  through  the  earth's 
axis,  and  the  second  through  the  point. 

It  is  with  the  Astronomical  Azimuth  only  that  we  are  at 
present  concerned.  The  azimuth  may  be  reckoned  from 
either  the  north  or  south  point  of  the  horizon.  For  astro- 
nomical purposes  it  is  usually  reckoned  from  the  south  point 
towards  the  west  from  zero  to  360°.  In  determining  the 
azimuth  of  a  point  on  the  earth's  surface  it  is  more  conven- 
ient to  use  stars  near  the  north  pole  of  the  heavens;  conse- 
quently for  geodetic  purposes  the  azimuth  is  generally 


520 


PRACTICAL  ASTRONOMY. 


§295- 


FIG.  580. 


§  295-  DETERMINATION  OF  AZIMUTH.  $21 

reckoned  from  the  north  point.  For  the  sake  of  uniformity 
we  shall  in  this  chapter  always  suppose  the  azimuth  reckoned 
from  the  north  in  the  direction  N.,  E.,  S.,  W.  A  minus  azi- 
muth will  be  reckoned  from  north  towards  west. 

Extreme  accuracy  in  the  determination  of  azimuth  is  re- 
quired in  connection  with  the  geodetic  operations  of  primary 
triangulation.  The  principal  methods  employed  in  such 
cases  will  be  given,  when  it  will  be  shown  how  they  may  be 
abridged  where  a  less  degree  of  accuracy  is  demanded. 
There  is  a  variety  of  these  methods,  depending  on  the  form 
of  instrument  employed  and  the  position  of  the  stars  ob- 
served. The  instrument  will  be  either  the  theodolite,  used 
for  measuring1  horizontal  angles,  or  the  astronomical  transit. 
In  any  case  the  azimuth  of  the  point  is  determined  by  meas- 
uring instrumentality  the  difference  between  the  azimuth  of 
the  point  and  a  star.  The  azimuth  of  the  star  is  computed  by 
its  known  right  ascension  and  declination,  and  the  local  time 
and  latitude,  which  have  been  previously  determined ;  from 
these  data  we  have  the  azimuth  of  the  point. 

295.  The  Theodolite.  Figures  $8a  and  58^  show  two  forms 
of  instruments  used  on  the  U.  S.  Coast  Survey.  The  older 
form,  Fig.  58^,  has  a  horizontal  circle  from  20  to  30  inches 
in  diameter.  With  the  newer  instruments,  circles  from  12  to 
20  inches  are  considered  sufficiently  large,  as  such  circles 
can  now  be  graduated  much  more  accurately  than  formerly  ; 
the  instrument  can  therefore  be  made  more  compact  and 
portable,  a  matter  of  some  importance  in  the  field. 

The  horizontal  circle  is  commonly  divided  directly  to  5', 
these  spaces  being  subdivided  by  reading  microscopes 
directly  to  single  seconds,  and  by  estimation  to  tenths  of  a 
second.  Two  or  three  microscopes  are  used.  The  essential 
features  of  the  instruments  will  be  understood  from  the 
plates  without  further  description. 

For  secondary  azimuths  a  less  perfect  instrument  will  often 


522 


PRACTICAL  ASTRONOMY. 


§295- 


FIG.  58*. 


§296. 


DE  TERM  IN  A  TION  OS  A  ZIM  u  Til. 


523 


be  used.  For  magnetic  work  or  ordinary  land-surveying  a 
common  surveyor's  transit  with  5-  or  6-inch  circle  will  fre- 
quently be  employed.  It  is  perhaps  unnecessary  to  say  that 
the  instrument  must  be  carefully  adjusted  in  every  particular. 
296.  The  Signal.  For  observing  at  night  an  illuminated 
mark  is  required.  A  convenient  mark  is  a  square  wooden 
box  firmly  mounted  on  a  post  or  other  support,  the  light  of 


FIG. 


a  bull's-eye  lantern  being  thrown  through  a  small  hole  in  the 
front.  The  box  itself  may  be  painted  so  as  to  form  a  con- 
venient target  for  day  observation.  This  mark  must  be 
placed  far  enough  from  the  station  so  that  no  change  will  be 
required  in  the  sidereal  focus  of  the  telescope :  about  one 
mile  will  generally  be  'sufficient.  When  from  any  cause  a 
distant  mark  is  not  practicable  a  colli mating  telescope  may 
be  used ;  but  the  greatest  care  must  be  exercised  in  mount- 


524  PRACTICAL   ASTRONOMY.  §  298. 

ing  both  the  instrument  and  collimator  firmly,  piers  of  solid 
masonry  being  used  for  both. 

297.  Choice  of  Stars.     For  first-class  azimuths  only  close 
circumpolar  stars  will  be  used.     Preference  will  be  given  to 
the  four  circumpolar  stars  whose  places  are  given  in  the 
ephemeris,  viz.,  **,  tf,  and  A  Ursae  Minoris,  and   51   Cephei. 
Fig.   59  shows  their  relative   positions,   and    will   assist   in 
finding  the  smaller  ones  which  are  not  readily  distinguished 
with  the  naked  eye  unless  the  position  is  previously  known. 

298.  Method  of  Observing.     A  complete  series  of  observa- 
tions on  one  star  will  consist  of  ten  or  twelve  readings  on  the 
mark  and  about  the  same  number  on  the  star,  the  instrument 
being  reversed  about  the  middle  of  the  series.     The  follow- 
ing order  of  observation  is  recommended  : 

i st.  6  readings  on  the  mark. 

2d.   6  readings  on  the  star. 

3d.  Read  the  level. 

4th.  Reverse. 

5th.  Read  level. 

6th.  6  readings  on  the  star. 

/th.  6  readings  on  the  mark. 

If  more  than  one  series  is  taken  it  is  advisable  to  change 
the  position  of  the  horizontal  circle  so  as  to  bring  the  read- 
ings in  another  place,  in  order  to  eliminate  to  some  extent 
the  errors  of  graduation. 

Readings  are  sometimes  taken  on  the  star  directly,  and  on 
its  image  reflected  from  a  basin  of  mercury.  When  this  is 
done  reading  the  level  may  be  dispensed  with. 

By  the  process  above  described  we  have  a  carefully-exe- 
cuted measurement  of  the  difference  in  azimuth  between  the 
star  and  mark.  It  only  remains  to  compute  the  azimuth  of 
the  star,  when  we  shall  have  the  azimuth  of  the  mark. 


ERRORS  OF  COLLIMATION  AND  LEVEL. 


525 


Let  m  —  reading  of  circle  on  mark  ; 

s  —  reading  of  circle  on  star ; 
A  —  azimuth  of  mark  measured  from  north  towards 

east ; 

a  =  azimuth  of  star  measured  from  north  towards 
east. 


Then 


A  =  a  -f  (m  —  s). 


(487) 


Different  methods  of  computing  a  will  be  employed,  de- 
pending on  the  position  of  the  star  when  observed. 

Errors  of  Collimation  and  Level. 

299.  The  mark  and  star  being  at  different  altitudes  above 
the  horizon,  the  measured  difference  of  azimuth  will  be 
affected  by  an  error  of  collimation,  also  by  a  want  of  parallel- 
ism between  the  horizontal  axis  and  the  horizon. 

Other  theoretical  errors  of  the  instrument  we  need  not 
consider,  since  their  effect  may  be  made  inappreciable  by 
careful  adjustment. 

In   the   figure   let   NWSE   represent  the   horizon,  z  the 
zenith,  s  any  star,  w'  the  point 
where  the  horizontal  axis  pro- 
duced    pierces     the     celestial 
sphere. 

*b  is  the  inclination,  -f-  when 
west  end  of  axis  is  high ; 

*c,  error  of  collimation,  -f-  when 
thread  is  east  of  collima- 
tion axis ; 

x,  error  in  reading  of  horizon- 
tal circle  due  to  b  and  c. 


*  This  designation  is  sufficiently  general  for  our  purpose,  since  we  shall  only 
have  occasion  to  apply  it  to  stars  observed  near  the  pole. 


526  PRACTICAL   ASTRONOMY.  §  3OO. 

Then  in  the  triangle  sw'z,  sz  =  z  =  zenith  distance  of  star; 

zw   —  90°  —  b ;        w's  =  90°  +  c ;        w'zs  —  90°  +  x. 
Therefore  —  sin  c  =  sin  b  cos  z  —  cos  b  sin  #  sin  x. 


Or,  since  <:,  £,  and  x  will  be  very  small,  the  above  may  be 
written 

—  c  =  b  cos  z  —  x  sin  z ; 
from  which  *  =     ~     +          (487)' 


It  will  seldom  be  necessary  to  apply  the  correction  for 
collimation,  since  it  may  be  eliminated  by  observing  in 
both  positions  of  the  axis. 

If  the  mark  is  not  in  the  horizon  a  similar  correction  to 
readings  on  mark  will  be  required,  where,  of  course,  for  z  we 
shall  have  the  zenith  distance  of  the  mark. 


Azimuth  by  a  Circumpolar  Star  near  Elongation. 

300.  When  the  star  is  within  a  short  distance  of  elongation, 
either  east  or  west,  the  position  is  especially  favorable,  since 
the  motion  in  azimuth  then  is  very  slow.  Only  one  rending 
can  be  taken  at  elongation,  but  we  may  apply  a  correction 
to  the  readings  near  elongation  to  reduce  them  to  the  read- 
ing at  elongation. 

The  azimuth  and  hour-angle  of  the  star  at  elongation  are 


§  301- 


AZIMUTH  BY  A    CIRCUMPOLAR   STAR. 


527 


computed  by  considering  the  right-angle  triangle  formed  at 
this  instant  by  the  zenith,  pole,  and  star. 

Let  —  a*  and  te  be  the  azimuth  and  hour-  P 

angle  at  elongation  ; 
a,  d,  and  6,  the  right  ascension,  declina- 
tion, and  sidereal  time. 

Thenf 

—  sin  ae  =  cos  8  sec  <p\ 
cos  t   =  cot  3  tan  <; 


Chronometer  time  of  elongation  =  6  — 


The  chronometer  correction  should  be  known  within  about 
one  second,  and  may  be  determined  by  any  of  the  methods 
previously  given ;  or  the  theodolite  itself  may  be  used  for 
the  purpose,  either  as  a  transit  or  by  measuring  altitudes 
as  with  the  sextant,  provided  it  has  a  good  vertical  circle. 

301.  The  formulae  for  reducing  the  readings  to  elongation 
will  now  be  developed. 

Formulae  (121)  give  the  values  of  //  and  a  in  terms  of  3  and 
/  for  a  star  at  any  hour-angle.  Recollecting  that  we  now 
measure  the  azimuth  from  the  north  instead  of  the  south 
point,  these  equations  are 

(a)  cos  h  cos  a  =  sin  d  cos  cp  —  cos  d  sin  <p  cos  /; 

(b)  cos  h  sin  a  =  —  cos  d  sin  /. 


*  —  ae,  since  a  plus  value  of  the  hour-angle  te  corresponds  to  a  minus  azimuth. 

f  If  many  observations  of  the  same  star  are  to  be  made,  it  will  be  convenient 
to  prepare  in  advance  a  table  of  the  values  of  ae  and  0  extending  over  the  time 
during  which  it  is  intended  to  observe. 


528  PRACTICAL  ASTRONOMY.  §  3OI. 

At  elongation  we  have 

cos  8       sin  d  cos  te 

(c\  —  sin  ae  =  -      -  =  = =; 

cos  cp  sm  (p 

(d)  cos  ae  =  sin  #  sin  te. 

Multiplying  together  first  (a)  and  (c),  then  (b)  and  (d),  we 
'have 

(e)  —cos  h  cos  #  sin  ae  —  sin  tf  cos  d  —  sin  d  cos  d  cos  /  cos  te\ 

(f)  cos  ^  sin  a  cos  <ze  =  —  sin  d  cos  tf  sin  /  sin  te. 

Add  (/)  to  (e), 

—  cos  //  sin  (ae—a)  =       sin'tf  cos  tf — sin  tf  cos  tf  cos  (te—t). 

sin  #  cos  d 
From  this,  sm  (ae—a)  = —7 —  2  sin  -J-  (/.  —  /). 


The  computation  will  be  more  convenient  if  for  cos  h  we 
substitute  its  value  in  terms  of  ae  and  #,  viz., 

cos  h  —  —  cot  ae  cot  tf; 
and  therefore   sin  (ae  —  a)  =  tan  #e  sin2  d  2  sin2  J(/e  —  /).  (489) 

We  now  have  an  equation  which  gives  the  difference 
between  the  azimuth  at  elongation  and  at  any  hour-angle  t. 

As  this  will  only  be  used  for  stars  near  elongation,  and 
consequently  te  —  /,  a  small  quantity,  it  will  be  convenient 
to  expand  it  into  a  series,  viz., 

.  2  sin2 ±(te—f)       i  .  .   ,  „.. .  [2  sin2 £ (te  —  /)]*  „      , 

ae  —  a  =  tan  ae  sin  o  —  — h  —  (tan a«  sm2 Or  ^ : -, .        (490) 

sin  i  6 v  sm  I 


In  this  case      (a«  —  a)  =  sin"  x  [tan  ae  sin2  S  2  sin8  £(/e  —  /)]. 


§302.  AZIMUTH  BY  A    CIRCUMPOLAR   STAR. 

When  this  formula  is  applied  to  the  close  circumpolar 
stars,  sin*  d  differs  but  little  from  unity,  and  the  last  term 
will  in  all  practical  cases  be  inappreciable. 

We  have  therefore  the  simple  formula 

2  sin8  \(te  —  f)  . 

ae-a  =  tan  ae  -  ~-~    --  '•    •     •     •     (490 


302.  Correction  for  Inclination  of  Axis.  When  the  west  end 
of  the  axis  is  high  the  reading  of  the  horizontal  circle  will 
be  small;  therefore  the  correction  will  be  //#.$•. 

The  inclination  will  be  given  by  the  formula  derived  for 
transit  instrument,  (289)  : 


.    .    .    (492) 
Or  if  the  level  is  reversed  more  than  once, 

(493) 


Where  JFand  E  are  the  means  of  the  readings  of  the  east 
and  west  ends  respectively. 

The  effect  upon  the  reading  of  the  horizontal  circle  we 
have  by  equation  (487)!,  viz., 


Where  h  is  the  altitude  of  the  star. 

Such  a  correction  must  also  be  applied  to  the  reading  on 
mark  when  appreciable. 


530 


PRACTICAL   ASTRONOMY. 


§303. 


With  the  circumpolar  stars  observed  at  elongation  we  may 
write  tan  cp  for  tan  h.     Then  we  have 


Correction  for  level  =  6a  =  -  [  W  —  E\  tan  q>.  .     .    .     (494) 


303.  Correction  for  Diurnal  Aberra- 
tion. Suppose  at  the  instant  of  obser- 
vation the  point  from  which  observa- 
tion is  made  to  be  moving-  in  the 
direction  AB. 

Let  SA  be  the  true  direction  of  a 
ray  of  light  coming  from  a  star;  then 
in  consequence  of  aberration  the  star 
will  appear  in  the  direction  AS'. 

Let  AC  be  drawn  equal  to  the  dis- 
tance traversed  by  the 
ray  of  light  in  one  second 
-  F; 

AD,  the  distance  traversed  by 
the  point  on  the  earth's 
surface  in  one  second 


FIG.  62. 


Let  angle  SAB=^    S'AB=$f.    Then 
sin  z/5        v 


Then 


sinS 


or 


=  -77  sin  3: 


We  have  found,  equation  (286),  p  =  o".3i9  cos  (p. 
Therefore  4$  =  ".319  cos  q>  sin  3- (495) 


§303- 


AZIMUTH  BY  CIRCUMPOLAR   STARS. 


531 


This  gives  the  displacement  in  the  plane  determined  by 
the  direction  of  the  ray  of  light  and  the  direction  of  motion 
of  the  point  of  observation.  It  re- 
mains to  determine  its  effect  on  the 
star's  azimuth. 

In  Fig.  63  let  s  be  any  star,  NS 
the  meridian,  NESW  the  horizon. 
sA  is  drawn  perpendicular  to  the 
horizon,  and  therefore  equals  the 
altitude.  NA  equals  the  azimuth. 
The  angle  at  E  is  called  y. 

Since  the  point  occupied  by  the 
observer  is  moving  directly  towards 

the  east  point  of  the  horizon  at  the  instant  of  observation,^ 
will  be  equal  to  3". 

Then  the  right  triangle  sEA  gives  the  equations 


(a) 


cos  h  cos  a  =  sin  3  cos  y\ 
cos  h  sin  a  =  cos  3. 


We  require  the  effect  produced  on  a  by  a  small  change  in 
3;  therefore  we  differentiate  with  respect  to  h,  a,  and  3. 

—  cos  h  sin  a  da  —  sin  h  cos  a  dh  =  cos  3  cos  yd$  ; 
cos  h  cos  a  da  —  sin  h  sin  a  dh  =  —  sin  3  d$. 


Multiply  the  first  of  these  by  sin  a,  the  second  by  cos  a, 
subtract  to  eliminate  dh,  and  reduce  by  (a)  and  (&);  we  readily 
find 


da  =  -  -^- 


cos  a 


sin  3  cos  h 


532 


PRACTICAL   ASTRONOMY. 


§304. 


Substitute  for  d$  the  value  of  AS  given  by  (495),  and  recol- 
lect that  the  azimuth  is  reckoned  from  the  north;  we  have 


3a=  - 


319  cos  cp  cos  a 
cos  h 


.    (496) 


For  a  close  circumpolar  star  this  will  not  differ  appre- 
ciably from 


da=  ".319  cos  #. 


(497) 


This  will  be  added  algebraically  to  the  computed  azimuth 
of  the  star. 

304.  Formula  for  Azimuth  by  a  Circumpolar  Star  near  Elon- 
gation. 


sin  ae  —  cos  d  sec  cp\ 
cos  te  =  cot  d  tan  ?>; 


Chron.  time  =  a  ±  te  —  A6 


ae  —  a  —  tan  ae 


western 
eastern 

2  sin2  %(t  -  t] 
sin~F" 


Level  =  -\W ' -  E\  tan  q>\ 

Aberration  =  ".319  cos  a\ 

A  =  ae-{-(m—s)*—  level  +  aberration. 


(XXIV) 


m  =  reading  of  circle  on  mark;  s  =  reading  on  star. 


304- 


AZIMUTH  BY  CIRCUMPOLAR   STARS. 


533 


Example. 

1847,  October  I7th,  Polaris  was  observed  near  western  elongation  at  Aga- 
menticus,  York  County,  Maine,  with  one  of  the  30  inch  theodolites  of  the  Coast 
Survey,  as  follows:* 


No. 

Object 

Tel. 

Time    by 
Sidereal 
Chrono- 
meter. 

Azimuth  Circle. 

Level. 

A 

B 

C 

h.  m.    s. 

0          / 

d.       d. 

d.        d. 

d.        d. 

u  idiv.=o".97 

i 

Mark. 

R. 

6  30 

63  55 

39-  7     39-  o 

27.  5    27.  o 

27.  7     26.  5 

2 

33 

63  55 

41.  o    39.  7 

27.  o    28.  o 

26.  o     24.  3 

c  M. 

3 

34 

63  55 

41.  o    41.  o 

29.  8    29.  o 

26.  4     26.  3 

o  o 

4 

D. 

37 

243  55 

26.  2      28.  2 

16.  8     17.  o 

16.  8     13.  3 

It 

5 

39 

243  55 

25.  5     28.  o 

17.  o     17.  o 

16.  4     15.  2 

c  c 

6 

42 

243  55 

27.  o    29.  o 

19.  o     19.  o 

16.  2     14.  o 

o  2 

u         Level. 

i 

Star. 

D. 

6  47  12 

127  42 

68.  o    67.  o 

61.  5     63.  o 

64.  5     64.  3 

C               E.      W. 

2 

49  06 

127  42 

65.  o    65.  o 

63.  5     63.  2 

63.  i     60.  5 

2           44     62 

3 

5i  38 

127  42 

62.  8     62.  8 

57-  o    59-  8 

60.  0       58.  2 

u           63     44 

4 

I 

R. 

52    12  .5 

53  55  -5 
7  oo  54 

127  42 
127  42 
307  42 

58.0     58.0 
56.  o    57.  o 

48.2  48.7 

54.  o    52.  5 
51-1     52.0 
45.  2     45.  o 

55-  3     53-  5 
53-  o    52.  o 
47-  7     45-  8 

o 
^  o      43     63 
.0  d      64     43 

I 

2  25  .5 
4  01  .5 

3°7  42 
307  42 

48.  0       49.   2 

48.  o    48.  7 

43.   2       44.  2 

43-  °    44-  7 

45-0    44.8 
46.  8    45.  o 

w          46    62 

t                 fio        jfi 

9 

5  51 

3°7  42 

49.  o    49.  o 

44.  7     45.  o 

47.  9     46.  9 

t?          02     4^ 

10 

7  14-5 

307  42 

49-  2     50.  5 

44-8     44.8 

47.  2       46.  2 

.      43     63 

7 

Mark. 

R. 

7  16 

63  55 

40.  o    40.  o 

23.  o     25.  o 

26.  8    25.  2 

o          63     43 

8 

17 

63  55 

39-  7     39-  7 

23.  o     23.  o 

25.  7     24.  8 

S  rf  " 

9 

10 

D. 

18 
23 

63  55 
243  55 

38.  o    39.  o 
26.  o    26.  5 

21.  5       22.   7 

13.  7     14.  o 

25-  o    23.  8 
15.  o     14.  6 

o  o 

It 

ii 

24 

243  55 

26.  8    26.  8 

14-  5     14-  8 

15.  2     14.  o 

c  c 

12 

26 

243  55 

26.  7     27.  3 

14.  o     13.  o 

M.  5     13-  9 

u 

The  horizontal  circle  was  read  by  means  of  three  microscopes  designated 
A,  B,  C  respectively;  the  value  of  one  division  of  the  micrometer-head  cor- 
responding to  one  second  of  arc,  subject  to  the  correction  for  run.  The  circle 
being  graduated  directly  to  5',  if  five  revolutions  of  the  screw  exactly  cover  this 
space  there  is  no  correction  for  run;  otherwise  it  represents  the  excess  or 
deficiency. 

For  reducing  these  observations  we  have: 


Right  ascension  of  Polaris  —    a  =    ih    5™  32s. 96 

Declination  of  Polaris          =    S  =  88°  29'  54". 27 

Latitude  of  station  =    (p  =  43    13  25  .o 

Chronometer  correction       =  /70  =     —  im  51". 8 


534 


PRACTICAL  ASTRONOMY. 


§304 


We  first  compute  the  azimuth  and  time  of  elongation: 


cos  6  =  8.4183795 
cos  cp  =  9.8625407 
sin  ae  =  8.5558388 

ae  =  —  2°  3'  39". 21 
(ae  is  minus,  since  elongation  is  west.) 


cot  d  =  8.4185287 
tan  <p  =  9.9730531 

COS  te  =  8.3915818 

te  =  88°  35'  i7".8 
te  =    5"  54™  2l8.2 
a  =    i      5    33  -o 
0  =    6    59    54  .2 
4Q  =      —  i    51  .8 
Chronometer  time  of  elongation  =    7h    im  468.o 

In  the  table  which  follows,  the  column  marked  corrected  readings  is  the  mean 
of  the  readings  of  the  three  microscopes  corrected  for  run  when  necessary;  the 
remaining  columns  will  be  explained  by  referring  to  formulae  (XXIV). 


No. 

Position. 

Corrected 
Readings. 

/e-'. 

2SinH(/e-*) 

at  -  a. 

Reduced 
Readings. 

Means. 

sin  i" 

i 

R. 

63055'3i"-3 

2 

31    .1 

3 

32   -3 

4 

D. 

243  55  19   -8 

5 

19   .9 

6 

20    .8 

j 

D. 

127  42  64  .7 

+I4m34* 

4i6".s 

-  iS".o 

i27°42/49//.7 

2 

63   .4 

12     40 

315    .0 

"    -3 

52   .1 

3 

60  .1 

10      8 

201     .6 

7   -3 

52   .8 

4 

55   -2 

9    33  -5 

179  .4 

6   -5 

48   .7 

5 
6 

R. 

53   -5 
307  42  46   .8 

7    50-5 

+        52 

120    .7 

1    -5 

4   -3 
.1 

49   .2 
307  42  46   .7 

Level  —  .23 

7 

'    45   -7 

-        39-5 

.8 

.0 

45    -7 

8 

46   .0 

2      15-5 

10     .0 

•3 

45    -7 

9 

47    -1 

4      5 

32   -7 

I     .2 

45    -9 

10 

47    -1 

-5    28  .5 

58    -9 

—      2     .1 

45-   ° 

307  42  45   .80 
Level        .00 

I 

R. 

63  55  3°  .0 

29    .4 

9 

28    .4 

10 

D. 

243  55  18   .3 

ii 

18   .7 

12 

18   .3 

Mean  of  readings  on  mark  =  m  =  243°  55'  24". 86 

48  .03 
36  .83 
3  39  -2I 
8   57  -62 
+  .32 
8'  57"- 94 


Mean  of  readings  on  star     =   s  =  127    42 
m  —  s  =  116    12 

Azimuth  of  star  =  ae  —  —  2 

Azimuth  of  mark  A  =  114 

Diurnal  aberration 
Final  value  of  azimuth,  114 


§  305. 


STAR  AT  ANY  HOUR-ANGLE. 


535 


From  the  level  readings  we  have  — 

Direct. 

E  =  53-50 
^=53-00 

\\W-E\  =-.24 


Reverse. 
53.50 
53.50 


Azimuth  by  a  Circumpolar  Star  observed  at  any  Hour-angle. 

305.  This  method  differs  from  the  preceding  in  the  manner 
of  computing-  the  azimuth  of  the  star,  which  may  be  con- 
veniently done  by  either  of  three  methods. 

First.  By  the  fundamental  equations  (a)  and  (b\  Art.  (301), 
we  readily  find 

sin  /  , 

tan  a  = — -s : -.  .     .     (498) 

cos  q>  tan  o  —  sin  cp  cos  t 

Second.  We  may  apply  Napier's  analogies  to  the  triangle 
formed  by  the  zenith,  pole,  and  star,  viz., 


•    •    (499) 


Third.  By  expansion  into  series. 

In  equation  (498)  write  /  —  90°  —  tf.     Then 

sin  t  sin  p 
~  cos  <p  cos  p  —  sin  cp  cos  t  sin  /' 

a  and  p  being  small,  we  may  expand  tan  a,  sin  /,  cos  p  into 


536  PRACTICAL  ASTRONOMY.  §  305. 

series,  when  the  equation  becomes,  to  terms  of  the  third  order 
inclusive, 

is  =         __  sin  /(/  -  j/3)  _ 
"  cos  (p(i  —  i/)  —  sin  cp  cos  t(p  —  |/3)' 
or 

a  cos  cp—  —p  sin  t-\-ap  sin  (p  cos  /+J#/2cos  <p—  J^'cos  <p+-J/3sin  /. 

Solving  this  equation  for  a  by  approximations,  we  have  for 
the  first  approximation 

sin  / 


Or      —       ~~~ 


COS 


This  value  substituted  in  the  second  term  of  the  second  mem- 
ber of  the  above  equation  gives  for  a  second  approximation 


sn 


This  value  substituted  in  the  second,  third,  and  fourth  terms 
of  the  above  gives  finally 

a  =  —  /4-/2sin  i"tan  <pcos/+^3sin2i"[(i4-4tan2<p)cos2/—  tan9®]     .(500) 

cos<p|_  J 


For  Polaris  within  the  limits  of  the  United  States  the  term 
in/3  will  not  exceed  2"  ,  while  the  terms  neglected  will  not 
be  greater  than  o".i. 

For  a  close  circumpolar  star  observed  near  culmination 
this  formula  may  be  written 


a—— I  /4-/2sini'/tan(pcos/+£/3sinV/(i+3  tanV)  \'(S°l) 


sin  /  F 
cosq>\ 


The  corrections  for  level  reading  and  aberration  will  be  com- 
puted by  the  same  formulae  as  in  the  previous  case. 


§  306.          CORRECTION  FOR  SECOND  DIFFERENCES.  537 


Correction  of  the  Mean  Azimuth  for  Second  Differences. 

306.  In  applying  the  foregoing  method  to  a  series  of  ten 
or  more  readings  on  a  star  we  may  proceed  in  either  of  two 
ways :  first,  we  may  reduce  each  reading  separately,  com- 
puting the  azimuth  of  the  star  for  each  time  of  observation  ; 
or  second,  we  may  take  the  mean  of  the  readings  and  com- 
pute the  azimuth  for  the  mean  of  the  corresponding  times, 
applying  to  this  computed  azimuth  a  small  correction  for 
second  differences. 

The  first  method  involves  considerable  labor,  but  at  the 
same  time  the  individual  values  furnish  a  rough  check  on  the 
accuracy  of  the  work.  When  the  second  method  is  pre- 
ferred we  may  derive  the  expression  for  the  correction  as 
follows : 

Let  /j,  /a,  /3,  .  .  .  tn  =  the  observed  times ; 

alt  a»  #3,  .  .  .  an  =  the  corresponding  azimuths  of  the  star; 

— ~^          — -  =  /0=  the  mean  of  the  observed  times ; 
a0  =  the  azimuth  corresponding  to  t0. 
Let  At,  =  t,  -  /. ;         At,  =  t,  -  /„ ;  .  .  .  Atn  =  tn  -  /0. 
Then  \ye  have     At,  +  At^  -f-  .  .  .  -f-  Atn  =  o. 
We  may  now  write 

da  d*a 


=A*,)  =A*. 


~ 


538  PRACTICAL  ASTRONOMY.  §  306. 

The  mean  of  these  expressions  will  be 


_ 
n  *^  df  2  n 

The  quantities  At  will  be  expressed  in  time  :  multiplying 
by  15  to  reduce  to  arc,  and  also  multiplying  each  quantity 

of  the  form  (15^)"  by  sin  if>  ',  the  term  multiplied  by  -TT 
will  be 


.  (502) 


Or,  if  preferred,  this  term  may  be  computed  by  table  VIII  A, 
for,  since  the  quantities  At  will  be  small,  we  shall  have  prac- 
tically 


and  the  above  tertn  becomes 

I  _  2  sin3  \At 

-2—.  —  77-  ......   :    (503) 

n        sin  \" 
It  remains  to  determine  a  convenient  expression  for  —  ^-2-. 

Differentiating  equation  (£),  Art.  301,  with  respect  to  # 
and  /,  we  find 

d*a  _         tan  a  /cos*/  —  cosa<2\ 

"3?~=    +^77^      ^o7^      /'    '    '    '    (5°4) 

For  a  close  circumpolar  star  cos2^  differs  but  little  from 
unity,  so  that  we  shall  have  very  nearly 

d*a 

jf  =    ~  tan  a  .......     (505) 

*  It  will  be  seen  that  the  expression  which  we  have  derived  for  reducing  the 
reading  taken  near  elongation  to  the  reading  at  elongation  is  a  special  case  of 
this  same  form. 


§  307.          CORRECTION  FOR  SECOND  DIFFERENCES. 

We  therefore  have  for  the  mean  of  the  azimuths 


=  a0-  tan  *0  [6.73672] 


539 


(506) 


where,  as  usual,  the  quantity  in  brackets  is  a  logarithm,  and 
the  quantities  At  are  expressed  in  seconds  of  time. 

Example. 

307.     1848,  April  5.     Observations  on  Polaris  at  Dollar  Point,  Galveston 
Bay,  Texas.     Instrument,  1  8-inch  Troughton  &  Simms  theodolite. 

One  division  of  level  =  o''.82. 
cp  =  29°  26'    2".  6  ; 


AT-  —  i8.8. 


Azimuth  Circle. 

Level. 

Object. 

Position. 

Chronometer 
Time. 

A 

B 

c 

E. 

W. 

Mark. 

D. 

158°  50'  55" 

65" 

50" 

129 

7i.5 

R. 

51     20 

20 

00 

81 

"9 

126 

74 

Star. 

D. 

9h  3m  33"-5 

337    18  40 

35 

20 

83 

117 

4     47-5 

18   55 

55 

35 

6      7  .0 

18   75 

70 

55 

R. 

98      6.5 

»9   45 

55 

40 

9     24.0 

19   65 

75 

55 

10    23  .5 

20  20 

3<> 

10 

121.5 

79 

Mark. 

D. 

158    5°  55' 

65 

5° 

80 

1  20 

R. 

51     20 

15 

00 

121.5 

78 

77-5 

122 

The  reduction  is  now  as  follows  : 


Object. 

Position. 

Reduced 
Reading. 

Mean  of 
Readings. 

Chronometer. 

A/. 

A/2. 

Mark. 

D. 

158°  50'  56".7 

R. 

5i   13  -3 

Star. 

D. 

337    J8  31   -7 

9h    3m33*-5 

2IO*.2 

44184 

18   48   .3 

4     47  -5 

136  .2 

18523 

18   66   .7 

6      7  .0 

56.7 

3215 

R. 

19  46  .7 

8      6.5 

62.8 

3944 

19   65   .0 

9     24.0 

140.3 

19684 

337    20   20  .0 

337°  19'  26".4 

9    10    23  .5 

199.8 

39920 

Mark. 

D. 

158    50  56   .7 

R. 

Si    "    -7 

158    51     4   -6 

540 


PRACTICAL   ASTRONOMY. 


§307. 


Formula  (506): 

=  129470    log  =  5.1122 

log  —  =  9.2218 

ft 

Constant  log  =  6.7367 
tan  a  =  8.4092* 


log  correction  =  9.48oo« 
Correction  =  —  o".3 


Mean  of  times  =  gh  7m    3".  7 
AT-         —  i«.8 


a  —  i    4 
t  —  8h  2 


4  .7 


=  I20°44'i8".o 


The  azimuth  of   the    star  may  now  be  computed  either  by  equation  (498), 
(499),  or  (500).     We  shall  compute  it  by  each  method  for  illustration. 


Formula  (498)  is     tan  a  =  — 


sin  / 


cos  q>  tan  S  —  sin  q>  cos  /' 


<p  =  29°  26'    2". 6 
d  =  88    29  57  .83 


a  =  —  i°  28'  n".5 


cos  <p  =  9.9399792 

tan  d  =  1.5817575 

Sumi  =  1.5217367 

*  Zech  .0032688 

log  denom.  =.  1.5250055 

sin  /  =  9.9342512 

tan  a  =  8.4092457 


sin  <p  =  9.6914542 
cos  t  —  9.7o852i2« 
Suma  =  9-3999754* 
s\  —  J»  =  2.1217613 


Formulae  (499) :        tan  \(q  -\-  a)  = 


sn         — 


cot*/; 


d  =  88°  29'  57".83 
cp  =  29  26  2  .6 
-  <P=  59   3  55  -23 
—  <p)=  29  31  57  .61 
=  117  56  o  .43 


sin  =  9.6927762 


cos  =  9-9395566 


«£K 

58  58   0  .21 
60  22   9  .0 

28  32  20  .60 

cos  =  9.7122589 
cot  =  9.7549528 
\(q  -f-  a)  =  9.7354701  tai 

sin  =  9.9329140 

cot  =  9.7549528 

!*{?  —  «)  =  9.7615954 

d  ^=  — 

30  o  32  .09 

I  28  II  -5 

*  Addition  and  subtraction  logarithms. 


§  307.        AZIMUTH  BY  STAR  AT  ANY  HOUR-ANGLE.  541 

Formula  (500): 


sin  t  r 
[/-{-/  sin  i   tan  tpcos 


tan9?)  cos2/  - 


log/ 
log/2 
sin  i" 
tan  q> 
cos  / 

/ 

=  3.73257 
=  7-46514 
=  4.68557 
=  9.75I47 
=  9.70852* 

/  =  i 

log/3 
sin2  i" 
log* 

factor 
log  3d  term 

0  30'  2".  17 
5402".  17 

=  II-I977 
=    9-37II 
=    9-5229 

=    9-4401 
=    9.53I8 

tan2  q>  = 
log  4  = 
Sum  = 
log  (i  +  4  tan2  <p)  = 

rr>c2  / 

9-5029 
.6021 
0.1050 
.3567 
9.4170 
9-7737 
9.5029 
9-9372 
9.4401 

log  2d  term 

=  1.61070* 

Sum  = 
tan2  <p  = 
Zech  = 
log  factor  = 

2d  term  =  —  40".  80 
3d  term  =  +        -34 

Sum  =    5361  .71 
log  sum  =  3.72930 

sin  t  —  9  93425 
log  sec  <p  =    .06002 

log  a  =  3.72357» 


=  —  i°  28'  n".4 


For  computing  a  single  azimuth,  as  in  the  present  case,  formula  (498)  will  be 
preferred.  For  other  cases,  where  a  larger  number  of  values  are  required,  (499) 
and  (500)  will  sometimes  be  found  more  convenient. 

For  the  level  correction 


d  "  82 

-[  W  —  £]  tan  q>  =  -^—[97.56  —  102.44]  tan 


=  —  2.00  X  tan  <p  =  —  l".l3. 


Mean  reading  on  star  -|-  level  correction  =  337°  19'  25".  3  =  s. 

Mean  reading  on  mark  =  158    51     4  .6  =  m1 

Azimuth  of  star  +  correction  for  2  At*  -\-  aberration  =  —  i    28   10  .9  =  a. 
Azimuth  of  mark  =  a  -\-  (m  —  s)  =  180      3  28  .4  =  A. 


The  aberration,  as  before,  is  given  by  the  formula  ".32  cos  a. 


542  PRACTICAL   ASTRONOMY.  §  308. 

Conditions  favorable  to  Accuracy. 
308.   Reckoning  the  azimuth  from  the  north  point  equations  (121)  become, 

(a)  cos  h  cos  a  =  sin  <5  cos  cp  —  cos  8  sin  q>  cos  /  ; 

(b)  cos  h  sin  a  =  —  cos  d  sin  /  ; 

(c)  sin  //  =  sin  d  sin  <p  +  cos  8  cos  <p  cos  /. 

Also  from  the  triangle  whose  vertices  are  the  zenith,  polfj  and  star, 

(d)  sin  q  sin  d  =  cos  a  sin  t  —  sin  a  cos  /  sin  <p  ; 
(<?)   sin  q  cos  5  =:  —  sin  a  cos  <p  ; 

(/)  cos  q  =  —  cos  a  cos  /  —  sin  a  sin  /  sin  <p  ; 

^  being  the  angle  at  the  star. 
Dividing  (a)  by  (b)  we  find 

(g)  sin  /  cot  a  =  —  tan  8  cos  q>  +  cos  /  sin  cp. 
Differentiating  with  respect  to  a  and  t,  and  reducing  by  (/), 
da  sin  a  cos 


sin/ 


(5°7) 


This  reduces  to  zero  when  q  =  90°  ;  a  condition  possible  with  any  star  whose 
declination  is  greater  than  (p. 

With  a  close  circumpolar  star  at  elongation,  /  will  at  the  same  time  be  near  90" 
or  270°,  and  sin  a  will  be  small  ;  this  will  therefore  give  the  most  favorable  con- 
dition when  small  errors  in  t  are  to  be  apprehended. 

Differentiating  (g)  with  re&pect  to  a  and  d  and  reducing  by  (b}  and  (e), 

da  _        cos  cp  sin  a  _  sin  q 

d8  ~        cos  h  cos  8  ~~  cos~^   '     •••••••     (5°9} 

Differentiating  with  respect  to  (a)  and  (q>), 

—  =  tan  /^  sin  a    ........  '.     .     (510) 

Both  (509)  and  (510)  vanish  when  the  star  is  on  the  meridian  approaching  near 
maxima  values  for  a  circumpolar  star  at  elongation,  but  as  they  have  different 
signs  on  opposite  sides  of  the  meridian  they  will  vanish  from  the  mean  of  two 
determinations  arranged  symmetrically  with  respect  to  the  meridian. 

It  therefore  appears  that  the  azimuth  will  be  practically  free  from  the  effects  of 
small  errors  in  5,  /,  and  (p  if  it  is  determined  from  circumpolar  stars  observed  an 
equal  number  of  times  at  both  eastern  and  western  elongation. 

For  a  more  elaborate  treatment  of  this  subject  Craig's  Treatise  on  Azimuth  may 
be  consulted. 


§  309.     AZIMUTH    WHEN  THE    TIME  IS  NOT  KNOWN.         543 

Azimuth  by  the  Sun  or  a  Star  at  any  Hour-angle,  the  Time  not 

being  Known. 

309.  In  determining  azimuths  for  the  ordinary  purposes 
of  land-surveying  or  for  magnetic  work  extreme  accuracy  is 
not  required.  In  such  cases  it  may  be  derived  without  a 
knowledge  of  the  local  time  by  using  a  theodolite  and  read- 
ing both  horizontal  and  vertical  circles. 

Either  a  star  or  the  sun  may  be  employed ;  in  the  latter 
case  the  threads  are  placed  tangent  to  the  limbs  and  a  correc- 
tion for  semidiameter  applied.  The  vertical  thread  is  placed 
alternately  tangent  to  the  first  and  second  limbs,  and  the 
horizontal  thread  tangent  to  the  upper  and  lower  limbs.  If 
the  observations  are  arranged  symmetrically  with  respect  to 
the  limbs  the  semidiameter  will  disappear  from  the  mean. 

The  azimuth  of  the  star  is  computed  as  follows : 

The  last  of  equations  (113),  substituting  90°  —  z  for  h, 
and  recollecting  that  the  azimuth  is  reckoned  from  the  north 
point,  is 

sin  d  =  cos  z  sin  (p  -+-  sin  z  cos  cp  cos  a. 

d  and  (p  are  known ;  z  is  the  zenith  distance  measured  as  in- 
dicated, and  corrected  for  refraction,  and,  when  the  sun  is 
employed,  for  parallax.  We  therefore  solve  the  equation 
for  a. 

Writing  cos  a  —  I  —  2  sin2  \a,  then  cos  a  =  —  i  -|-  2  cos3  \a, 
we  find  by  a  familiar  reduction 


sin  COS     *         >  sn 


sn  z  cos  cp 


iin  \a  =A     - 

cosia=      / 

y 

/cos  ip  +  cp  -f-  6}  sin  %(z  +  <p 
~  V  cos  \(z  -  cp  -  d)  sin  40  -  cp 


sin  }(«  —  cp  -\-3)  cos  %(z  —  cp  —  d) 
sin  z  cos  <p 

*5 


544  PRACTICAL  ASTRONOMY.  §  310. 

The  azimuth  of  the  star  may  be  computed  by  either  of 
these  formulae,  the  last  being  most  accurate.  As  this  method 
will  not  be  employed  when  extreme  accuracy  is  required 
this  consideration  will  have  less  weight  than  in  other  cases. 

When  the  sun  is  employed  the  correction  for  semidiame- 
ter  is  obtained  as  follows  : 

Let  5  =  the  sun's  semidiameter  taken  from  the  ephemeris. 

Then  from  the  right-angle  triangle  formed 
by  the  great  circles  joining  the  zenith,  cen- 
tre, and  limb  of  the  sun  we  have,  calling  the 
angle  at  the  zenith  da, 

sin  5  =  sin  z .  sin  da, 

or          *«  =  ±  si-  •  •  •    <5I2> 

the  proper  algebraic  sign  being  obvious. 
If  the  time  is  also  required,  we  derive  it  from  the  meas- 
ured altitudes  by  the  method  of  Articles  (124)  and  (125). 

Conditions  favorable  to  Accuracy. 

310.  In  order  to  investigate  the  effect  upon  the  azimuth  of  small  errors  in 
assumed  latitude  and  zenith  distance  we  resume  the  fundamental  equation 

sin  8  =  cos  z  sin  tp  -\-  sin  z  cos  cp  cos  a. 

Differentiating  first  with  respect  to  a  and  z,  then  with  respect  to  a  and  cp,  we 
have 

dza  =  [—  tan  tp  cosec  a  -f-  cot  z  cot  a\dz  ;  )  ,      •. 

d$a  =  [—  tan  tp  cot  a  -j-  cot  z  cosec  a~\dcp.  ; 

The  coefficients  of  both  dz  and  dtp  diminish  as  a  and  z  approach  90°;  also  the 
coefficients  have  opposite  signs  for  a  —  90°  and  a  =  270°.  Therefore  by  select- 
ing stars  which  cross  the  prime  vertical  at  as  low  altitudes  as  may  be  consistent 
with  good  definition,  and  observing  at  about  the  same  distance  from  the  merid- 
ian east  and  west,  the  best  results  will  be  obtained. 

When  the  sun  is  used  it  should  be  observed  as  near  the  prime  vertical  as 
possible,  east  and  west. 

When  an  ordinary  surveyor's  theodolite  is  used  there  will  be  no  provision  for 


§3H.     AZIMUTH   WHEN   THE    TIME  IS  NOT  KNOWN.         545 


illuminating  the  field  ;  this  may,  however,  be  done  by  a  bull's-eye  lantern  held  in 
front  and  a  little  to  one  side  of  the  object-glass. 

Example. 

311.  Station,  Capital,  Washington,  D.  C. 

Sun  near  prime  vertical,  August  15,  A.M.,  1856.     Observer,  Charles  A.  Schott. 
Instrument,  5-inch  theodolite.  Longitude  5h  8m  Is  west  of  Greenwich. 


Chronom- 
eter* 
Time. 

Horizontal  Circle. 

Vertical  Circle. 

A 

B 

A 

B 

O  's  upper  and  first  limb.    Telescope  D. 

5h    2m  S38 
5     34 

6     55-5 

25°  24'  30" 
25    50  45 
26     4  30 

205°  24'  30" 
205    51   30 
206      5   15 

61°  56'    o" 
61    24  30 
61      8  45 

6i°56'    o" 
61    25     o 
61      9  30 

O  's  lower  and  second  limb.    Telescope  R. 

5      9     12 
10    32 
ii     42 

205    54   15 
206      7    15 
206    1  8   30 

25    54  oo 
26      6  45 
26    18   15 

61    19  30 
61      4  oo 
60    50  oo 

61    18  30 
61      3     o 
60    49  45 

Thermometer  73°. 


Barometer  30  inches 


We  also  have  <p  =  38°  53'  18"     Mean  chronometer  time*  =  5h  7m  48*.! 

d  =  13    55  33  Horizontal  circle  =  25 

Sun's  eq.  parallax  it  =  8". 5  Vertical  circle  =  61 

^  Refraction  =  r  =  -|- 

Parallax 
Corrected  zenith  dist.  =  61 


1702 
141  -7 
-  7  -4 

i8'36" 


We  compute  azimuth  of  star  by  the  last  of  (511) : 

i(*  +  <P  +  S)  =  57°    3:  44"  cos  =  9.73538 

i(z  -}-  cp  —  8}  =  43      8   ii  sin  =  9.83489 

%(z  —  (p  —  8)  =    4    14  53  sec  =    .00120 

-£(«  —  <p  -j-  5)  =  18    10  26  cosec  =    .50598 


tan 


•07745 
=    .03872.5 


\*  =  47°  33'    3"-o 

«  =  95      6     7 

Hor.  circle  =  25    56  40 

290    50  33  =  Reading  of  circle  for  north  point. 

*  A  sidereal  chronometer  was  used.    The  time  is  only  required  for  taking  8  from  the  ephem- 
eris  and  need  not  be  very  exact.     When  a  star  is  used  no  record  of  the  time  is  required. 


546  PRACTICAL  ASTRONOMY.  §313- 


Azimuth  by  the  Transit  Instrument. 

312.  It  has  already  been  shown,  in  connection  with  the 
general  theory  of  the  transit  instrument,  how  the  azimuth  of 
the  line  of  collimation  is  determined,  either  by  special  obser- 
vations made  for  this  purpose  or  from  a  series  of  transits  re- 
duced by  least  squares.     If  now  the  direction  of  this  line  is 
fixed  by  a  meridian  mark,  we  have  the  azimuth  of  the  mark. 
Such  a  determination,  though  not  of  the  highest  order  of  ac- 
curacy, is  sufficient  for  many  purposes. 

When  the  greatest  precision  is  required,  the  telescope 
must  be  provided  with  an  eye-piece  micrometer  moving  a 
vertical  thread.  The  instrument  will  generally  be  mounted 
either  in  the  meridian  or  in  the  vertical  plane  of  a  circum- 
polar  star  at  elongation. 

313.  Azimuth  by  a  Close  Circumpolar  Star  near  Culmination. 
The  instrument  is  set  up  and  adjusted  as  already  explained 
in  Articles  166-9.     The  mark  whose  azimuth  is  to  be  deter- 
mined must  be  placed  so  near  the  meridian  that  it  may  be 
well  observed  without  changing  the  azimuttuof  the  instru- 
ment.    In  positions  where  a  distant  meridian  mark  is  not 
available  a  collimating  telescope  may  be  used,  in  which  case 
the  firmest  possible  mounting  will  be  required  for  both  tran- 
sit and  collimator. 

The  observations  will  be  made  as  follows :  A  short  time 
before  the  star's  culmination  the  telescope  is  directed  to  the 
mark  and  a  series  of  readings  taken  with  the  micrometer, 
both  in  direct  and  reverse  position  of  the  instrument.  The 
level  is  then  read  and  a  series  of  transits  observed  over  the 
micrometer-thread,  which  is  moved  forward  successively  one 
turn  or  less.  The  instrument  may  be  reversed  or  not  at  the 
middle  of  the  series.  The  level  is  again  read  and  a  series 


§31$-      AZIMUTH  BY   THE    TRANSIT  INSTRUMENT.  547 

of  readings  on  the  mark  taken.  Transits  of  zenith  and  equa- 
torial stars  will  also  be  observed  for  determining  the  clock 
correction. 

314.  Method  of  Reduction.  The  value  of  one  revolution  of 
the  micrometer-screw  is  required.  If  not  previously  known 
this  may  be  derived  from  the  observed  transits  of  the  star, 
by,  the  same  method  used  for  determining  the  equatorial 
intervals  of  the  transit-threads,  viz.: 

Let  /  =  the  interval  of  time  required  for  the  star  to  pass 
over  the  space  corresponding  to  one  revolution 
of  the  screw. 


Then,  eq.  (291),        R  =  i5/cos  3  Vcos  /.  .....     (514)' 

Vcos  /  being  taken  from  table  Art.  174  when  it  differs 
appreciably  from  unity.  R,  the  value  of  one  revolution,  will 
be  expressed  in  seconds  of  arc. 

The  collimation  constant  may  be  derived  either  from  the 
transits  of  the  star,  the  instrument  being  reversed  at  the 
middle  of  the  series,  or  by  means  of  the  readings  on  the  mark 
in  the  two  positions  as  explained  in  Art.  182. 

When  the  transits  of  the  star  are  used  for  the  purpose  the 
formula  for  c  is  (see  Art.  185) 


c  =  $(T-T)  cos  8+%(Tf-  T)6Tcos  3  +  \(V  -  b)  cos  (>-#). 

It  is  well  to  derive  c  from  both  the  star  and   mark,  the  two 
determinations  mutually  checking  each  other. 

315.  The  mean  of  the  observed  times  must  next  be  re- 
duced to  the  time  over  the  line  of  coliimation  of  the  telescope. 


548  PRACTICAL   ASTRONOMY.  §  315. 

Let  rlt  ra,  .  .  .  rm  =  the  successive  readings  of  the  micro- 
meter; 

/0  /a,  .  .  .  tm  =  chronometer  times  of  observation; 
rc  and  /c  =  micrometer  reading  and  time  for  line  of 
collimation. 


' o   — 


m  m 


Then,  from  (291),,  tc-  t,   =  7?^-=-^  sec  *  Vsec  (/0  -  *0).  (515) 

The  factor  Vsec  (tc  —  /0)  is  taken  from  the  table  Art.  174 
if  it  differs  appreciably  from  unity.  We  thus  have  T,  the 
chronometer  time  of  transit  over  the  line  of  collimation. 

Then,  equations  (284),  (285),  (287), 

a  =  T+  AT  +  Aa  +  Bb  +  C(c  —  s.O2i  cos  <?);* 
in  which  A  =  sin  (cp—  #)sec  tf,  #=cos  (cp—  (^)sec^,  C=sec  $. 
Let      r  =  <x-[T+4T  +  £l>+C(c-*.02icos<p)']',  (516) 
that  is,  the  algebraic  sum  of  the  known  terms. 

Then  a 


is  the  expression  for  the  azimuth  of  the  star  in  seconds  of  arc. 
It  will,  however,  be  remembered  that  in  the  theory  of  the 

*  If  the  mean  of  the  times  has  been  reduced  to  the  line  of  collimation  as 
supposed  above,  c  will  be  zero;  if  not,  c  =  tc  —  to. 


315.      AZIMUTH  BY   THE    TRANSIT  INSTRUMENT. 


549 


transit  instrument  where  the  above  formula  is  derived,  a  is 
considered  plus  when  the  south  end  of  the  telescope  devi- 
ates to  the  east.  For  present  purposes,  therefore,  the  alge- 
braic sign  must  be  reversed,  giving  for  azimuth  of  star 


a'  =  — 


(5i8) 


The  azimuth  of  the  mark  then  follows  at  once  from  the  di£ 
ference  between  the  micrometer  readings  on  the  mark  and 
star. 

By  observing  the  same  star  at  both  upper  and  lower  cul- 
mination the  effect  of  any  constant  error  in  the  right  ascen- 
sion or  clock  correction  will  be  eliminated  from  the  mean. 


EXAMPLE. 

S  Ursa  Minoris  at  Lower  Culmination.  51  Cephei  at  Upper  Culmination. 

1882,  March  20.  Instrument,  Simms  Transit  C.  S.  No.  8. 


Chro- 

MARK. 

Chro- 

5 URS;E  MINORIS. 

LEVEL. 

51  CEPHEI. 

LEVEL. 

nome- 

nome- 

tet. 

Lamp 
E. 

Lamp 
W. 

ter. 

Micro- 
meter. 

Chro- 
nometer. 

E. 

W. 

Micro- 
meter. 

Chro- 
nometer. 

E. 

W. 

5h  20m 

18.760 

12.670 

5h  40™ 

18.22 

6hl8m44s 

13.22 

6h27m348 

18.760 

12.665 

17.72 

19    II 

13.72 

28     8 

18.750 

12.670 

17.22 

*9   37  «5 

49.8 

48.1 

14.22 

28  40  -5 

38.0 

63.0 

18.760 

12.665 

16.72 

20     4  .5 

35-o 

63.0 

14.72 

29   IS 

53-5 

49-o 

18.750 

12.675 

T.6.2* 

20   31  .5 

50.8 

48.2 

15-22 

29  48 

39-o 

63-5 

18.750 

12.675 

I5-72 

20   59 

36.3 

63.0 

15-72 

30     22 

53-5 

49.0 

18.751 

12.670 

15-22 

21     25    .5 

16.22 

30   54 

18.751 

12.672 

14.72 

21     52 

16.72 

31   29 

18.758 

12.670 

14.22 

22     19 

17.22 

32      i-5 

5  h  3om 

18.750 

12.665 

5h  50™ 

I3-72 

22     46 

17.72 

S2    36 

13.22 

6h23mI3B 

18.22 

6h33raI0. 

Means  18.754  12.670 


15.72      6b20m588.46  42.98  55.58         15-72    6h3om2ii.64  46.00  56.12 


cp  =  29°  7'  30" 


S  Ursae  Minoris. 
d  =  93°  24'  24" 
a  =    6h  20™   58.6i 


51  Cephei. 

S  =  87°  15'  33" 
a  =    6h  29m338.i5 


550 


PRACTICAL   ASTRONOMY. 


3I5. 


By  the  foregoing  formulae  we  compute — 

5  Ursae  Minoris. 
A  =  +  15.16 
B=-  7-30 
C=  -  16.83 
b  =  -f-  6".30=o8.42 


51  Cephei. 
A  =  -  17.76 
B  =  -}-  ii  -04 

C  —  -J-  20  .91 

£  =  -f-  5". 06  =  o".337 


We  now  derive  the  value  of  the  micrometer-screw  from  the  observed  tran- 
sits of  each  star,  as  follows:  Subtracting  in  each  case  the  first  time  from  the 
seventh,  the  second  from  the  eighth,  etc.,  we  have  the  following  values: 


6  Ursae  Minoris. 


log/—  1.73078 

log  15  =  1.17609 

cos  d  =  8.77395 

log  R  =  1.68082 
R  =  47.95 


3turns  =2m  41". 4 

I  turn  =       53  .80  =  I 


5f  Cephei. 


log  /  =  1.82607 
log  15  =  1.17609 
cos  d  =  8.67961 

log  R  —  1.68177 

R  —  48.06 
3  turns  =  3m  21" 
•i  turn    =        67  .o  =  7  Mean  R  =  48".oo 


The  mean  of  the  readings  on  the  mark  E.  and  W.  gives  rc  =  15.712.    There- 
fore, by  formulae  (515),  (516),  and  (518) — 


5  Ursae  Minoris. 

51  Cephei. 

Observed 

time 

=  6h  20m 

58s. 

46 

6h 

30m  2  1" 

.64 

tc 

—  to 

=  + 

42 

— 

•  54 

T 

=  6h  20m 

58'. 

88 

6" 

3Om  21s 

.10 

AT 

__  _ 

5i  - 

30 

— 

51 

30 

bB 

=  — 

3  • 

07 

+ 

3 

•  73 

*Cc' 

=  + 

31 

- 

.38 

a 

=  6    20 

5  • 

61 

6 

29     33 

.15 

r 

=  + 

B 

79 

r  = 

o 

oo 

a' 

~    — 

o". 

78 

a1  = 

,00 

*  c  is  =  o,  since  we  have  reduced  the  times  to  the  axis  of  collimation.    Therefore 

C1   —   —  8.02I  COS  <f> 

=  -  .018. 


§316.      AZIMUTH  BY   THE    TRANSIT  INSTRUMENT,  551 

Mark  west  of  collimation  axis  3.042  revolutions  =       2'  26". 02 
Mean  value  of  a'  =  —  .39 

Azimuth  of  mark  =  —  2   25   .63 

316.  If  the  telescope  is  not  provided  with  an  eye-piece  micrometer,  the  azi-' 
muth  screw  at  the  end  of  the  axis  may  be  employed  (see  description  of  instru- 
ment, Art.  158).  The  mark  in  this  case' must  be  quite  near  the  meridian,  as  the 
range  of  the  screw  is  small.  The  method  of  observing  is  the  same  as  that  de- 
scribed in  the  last  article. 

Determination  of  the  Value  of  the  Screw.  For  this  purpose  a  series  of  transits 
of  a  circumpolar  star  near  culmination  will  be  observed,  extending  over  the  en- 
tire available  range  of  the  screw.  It  will  be  as  well  not  to  extend  it  to  the  ex- 
treme limit  in  either  direction. 

Let  M  —  the  micrometer  reading  at  any  observed  time  /; 

Ms  =  the  micrometer  reading  at  time  of  culmination.  A>; 
1?  —  the  value  of  one  revolution  of  screw. 

Then  since  the  screw  moves  the  instrument  in  azimuth,  we  have,  by  (517), 

* 

R(M  -  M0)  =  ± 

where  r  —  /  —  to. 

This  is  a  little  more  accurately  written 


R(M  -  M$  sin  i"  =  —  sin  (isr), 

A 

or  R(M  -  M,)  =  ~[i$r  -  $(IST?  sin2  1"]; 


-  KIS  sin  i")»r«]  .....     (519; 


Where  the  log  £(15  sin  i")2  =  0.94518  —  10,  and  the  quantity  £(15  sin  i")8r8 
may  be  taken  from  the  table  Art.  275.  When  this  correction  is  appreciable  it 
will  be  convenient  to  apply  it  directly  to  the  observed  times,  when  we  shall 
have  these  times  reduced  to  what  they  would  have  been  if  the  star  had  moved 
uniformly  in  a  great  circle.  The  method  of  combining  these  reduced  times  is 
the  same  as  that  illustrated  in  the  preceding  article. 


552 


PRACTICAL  ASTRONOMY. 


317. 


EXAMPLE. 

8  Ursa  Minoris  near  lower  culmination,  February  5,  li 
Chronometer  time  of  lower  culmination,  6h  isin  48* 


Micr. 

Chron.  time 

Time  from 
culmina- 
tion. 

Red'n. 

Red'd  time. 

Time  of  3  turns. 

t. 

h.  nt.    s. 

fn. 

s. 

h.  m.    s. 

t.         t. 

m.      s. 

21.  0 

5  55  57 

19.9 

+  i.S 

5  55  58.5 

21    to  18 

g    62.7 

20.5 

57  40 

18.1 

i.i 

57  4I-I 

20.5      17.5 

9     59-5 

2O.  O 

59  23-5 

16.4 

0.8 

59  24.3 

20             I7 

9     55-7 

19  5 

6  01  02.5 

14.8 

0.6 

6  01  03.1 

19-5      16.5 

9     56-9 

19.0 

02    41.5 

13-1 

0.4 

02  41.9 

19         16 

9     57-1 

18.5 
18.0 

04    21.0 

06  01  .0 

"•5 
9.8 

0.3 

0    2 

04  21.3 
06  01.2 

18.5      15-5 
18         15 

9     5i-7 

9     55-8 

17-5 

07    40.5 

8.1 

O.  I 

07  40.6 

17.0 

09  20  o 

\     6.5 

0.0 

09  20.  o 

Mean 

9     57-°7 

21 

II    00.0 

4.8 

o.o 

11    00.0 

IO.O 

12    39.0 

3-2 

o.o 

12    39.0 

15-5 

14    13.0 

1.6 

0.0 

14    13-0 

15.0 

15  57-o 

O.I 

o.o 

'5  57-o 

Time  of  three  revolutions,  597s. 07 
One  revolution        =  r  =  199". o 


=  198". 2 


log  =  2.29885 
log  15  =  1.17609 

log—  =  8.82216 

A 
log  R  =  2.29710 


Star's  declination  =  d  =  93°  23'  48" 
Latitude  =  q>  =  30   13    54 

The  computation  of  the  azimuth  of  the  star  at  the  mean  of  the  observed  times, 
and  the  determination  of  the  azimuth  of  the  mark  from  the  combination  of  the 
readings  on  star  and  on  mark,  will  require  no  further  illustration. 


Azimuth  by  Circumpolar  Star  at  any  Hour-angle. 

317.  When  extreme  accuracy  is  required  the  instrument 
must  be  provided  with  an  eye-piece  micrometer.  The  mark, 
of  course,  must  be  near  the  line  of  collimation.  The  method 
of  observing  will  be  the  same  as  with  the  theodolite,  Art. 
298,  except  that  the  readings  are  made  with  the  micrometer. 

If  there  is  no  eye-piece  micrometer  the  azimuth-screw  may 


§3I/-      AZIMUTH  BY   THE    TRANSIT  INSTRUMENT.  553 

be  used,  in  which  case  the  reduction  will  be  precisely  the 
same  as  that  given  for  the  theodolite,  formulae  (XXIV),  Art. 

304. 

When  the  micrometer  is  em- 
ployed the  reduction  will  be  as 
follows : 

In  the  figure  NESW  repre- 
sents the  horizon,  P  the  pole,  s  the 
star,  Z  the  zenith,  ^  the  mark,  CZ 
the  direction  of  the  line  of  collima- 
tion,  w'  the  point  where  the  west 
end  of  axis  pierces  the  celestial 
sphere. 

FIG.  65. 

Let  M0  =  micrometer  reading  on  line  of  collimation ; 
M  —  micrometer  reading  on  star; 
M'  =  micrometer  reading  on  mark ; 
R    =  value  of  one  revolution  of  screw ; 
b    —  elevation  of  west  end  of  axis. 

Then  from  the  micrometer  and  level  readings  we  require  the 
expression  for  the  difference  in  azimuth  of  s  and  /*. 


Let 


Then  from  figure, 


R(M- 

R(M'  -  M,)  =  mr. 


Then  if          a  —  azimuth  of  star,          a'  —  azimuth  of  mark, 
a  —  a'  =  a,  —  #/  =  required  difference  of  azimuth. 


554  PRACTICAL   ASTRONOMY.  §3lS. 

From  triangle  w'zs, 

—  sin  m  =  sin  b  cos  z  —  cos  b  sin  z  sin  at. 
From  triangle  ze/,sr;w, 

—  sin  m'  =  sin  #  cos  z'  —  cos  £  sin  z'  sin  #/. 

« 

m,  m',  b,  alt  and  0/'will  always  be  small  quantities ;  therefore 
the  above  equations  may  be  written 

—  m   =  b  cos  z   —  a1   sin  z ; 

—  m'  =  b  co&  z'  —  a{  sin  z'. 

From  these  equations  we  obtain 

,          m  m'          ,  sin  (z'  —  z) 

a.  —  a,    =  - 7  4-  b  - — - — : 7-.      .     (1520) 

sin  z       sin  z  sin  z  sin  z 

The  micrometer  reading  is  supposed  to  increase  with  the 
azimuth ;  if  the  opposite  is  the  case  the  signs  of  m  and  m' 
will  be  changed. 

b  includes  the  correction  for  inequality  of  pivots ;  also  for 
flexure,  if  the  instrument  is  of  the  form  shown  in  Fig.  28. 
(See  Art.  192.)  Thus  the  complete  expression  for  b  is 

b  =  d-(W-  E)+p+f.     ....    (521) 

# 

p  is  the  correction  for  inequality  of  pivots,  and  /"the  flexure. 

The  azimuth  of  the  star  being  computed  by  any  of  the 
methods  before  given,  we  have  by  (520)  the  required  azimuth 
of  the  mark. 

318.  A  Circumpolar  Star  near  Elongation.  It  will  be  best 
when  practicable  to  observe  the  stars  near  the  time  of  elon- 


§319-      AZIMUTH  BY   THE    TRANSIT  INSTRUMENT.  555 

gation.     The  readings  on  the  star  may  then  be  reduced  to 
the  reading  at  elongation  as  follows:     In  the  figure  let 

se  =  position    of    the   star    at    time   ' 

Te  =  elongation ; 
s  =  position  of  the  star  at  time  T. 

Then  sea  =  x  is  the  correction    re-    s 
quired  to  reduce  the  reading  at  s  to 
the  reading  at  elongation. 

From  the  right-angle  triangle  sPa,   we  have 

cos  (te  —  t)  •=  tan  3  cot  (3  +  x). 
From  this,  by  the  process  given  for  deriving  equation  (483) 

,  2  sin2  Ute  —  t) 
x  =  i  sin  23  -     -^77—^ (522) 


On  account  of  the  rapidity  and  accuracy  with  which  the 
micrometer  readings  may  be  made  several  sets  may  be  taken 
at  one  elongation  if  thought  desirable. 

Example. 

319.   In  Vol.  XXXVII,  Memoirs  Royal  Astronomical  Society,  Captain  Clarke 
gives  among  others  the  following  observation  of  Polaris  : 

Station  Findlay  Seat,  1868,  October  23. 
Position  E 

W  —  E    —  -       1.30  Latitude  g>  =  57°  34'  50". o 

M  —  MQ  =        580.19  Declination  5  —  88    36  34  .4 

M'  —  M0  =  —    77.01  Right  ascension  a  —    ih  nm  57". 46 

Sidereal  time  =  i8h  4im  30°.  n  Hour-angle  /  =  17    29    32  .65 

t  —  262°  23'  9".  75 
Zenith  dist.  of  mark  z   =    93      2 

We  also  have        One  division  of  level  =  d  =  i".8io 

One  division  of  microm.  screw  =  R  =    ".8345 
Inequality  of  pivots  p  =    ".650 

Flexure  y=3"-I7i 


556  PRACTICAL   ASTRONOMY.  §319- 

The  observations  given  are  the  means  of  a  series  taken  in  the  following 
order : 

ist.   Level. 

2d.    Mark. 

3d.    Direct  telescope  to  star  and  read  level. 

4th.  Three  readings  on  star. 

5th.  Level. 

6th.   Mark. 

7th.  Level. 

The  instrument  is  then  reversed  and  another  series  taken  in  the  same  order. 
The  level  reading  given  is  the  mean  of  the  four  above  indicated. 

We  shall  first  reduce  the  observation  by  computing  the  azimuth  of  the  star  at 
the  instant  of  observation. 

As  both  zenith  distance  and  azimuth  are  required,  equations  (II),  Art.  (65), 
may  be  employed.     These  equations  are  rewritten  here  for  convenience. 

tan  8 


tan  t\ 


Proof: 

sin  (<p  —  M)       cos  h  cos  a 

By  means  of  these  formulae  we  readily  find 

a  =    2°  33'  23".s8 
h  =  57    22   13  .38 

2  =  32    37  47 
By  formula  (521), 

b  =  -       .65  X  i".8i  -f  3".  171  +  ".650  =  2".645 
m  —  580.19  X  .8345  log  =  2.68500 

»*'  =  —  77-01  X  .8345  log  =  1.80798* 

m 

=  +  14  57'  .9* 


tail 

tan 

tan 
cos  M 

2U     .  , 

cos  t 
cos  M 

sin  (cp  —  M) 
cos  a 

tan  (tf>  —  M) 
cos  6  cos  t  - 

,  sin  (zf  —  z) 

b   .         .  -  /  =4-          4  .27 
.     sin  z  sin  z' 

a  —  a'  —       16     6  .55 

Azimuth  of  star  =  a  =  2    33  23  .58  This  still  requires  the  correction  for 

Azimuth  of  mark  a'  —  2°  17'  17".  03         diurnal  aberration,  viz.,  -|-  o".32  cos  a. 


§  32O.      AZIMUTH  BY   THE    TRANSIT  INSTRUMENT.  557 


320.  The  observations  of  the  foregoing  example  are  taken  too  far  from  elon- 
gation for  reduction  by  formula  (522),  but  they  will  serve  to  illustrate  the  method. 
We  compute  the  azimuth  and  time  of  elongation  by  the  formulae 


We  readily  find 


Then  by  (522), 


sin  ae  =  cos  S  sec  <p 
cos  te  =  cot  d  tan  <p 
Time  of  elongation  Te  =  a  —  te 

ae  =  2°  35'  39".!! 
Te  =  IQh  20m  43s.  1 3 

Time  of  observation  T  =  18    41     30  .11 
Te  -  T  =  te  -  t=         39     '3  -02 


log 


2  sin2  \(te  -  /) 


26  =  177°  13'  8".8 


sini" 


'-  —  3-47892 

sin  =  8.68589 
gi  =  9-69897 


log  x  —  1.86378 


Reduction  to  elongation  =  x  =    73". 08 

Micrometer  reading  on  star  m  =  484  .18 

Reading  at  elongation  =  m  -\-  x  =  557  .26 

m  _j_  x  now  takes  the  place  of  m  in  equation  (520).  When  the  observation  is 
within  a  few  minutes  of  elongation  we  take  for  «  the  zenith  distance  at  time  of 
elongation  ;  but  in  the  present  example  this  will  not  be  admissible.  Using  for 
2  the  value  derived  in  the  previous  reduction,  we  have 


m  -{-  x 

if  I3".48 
i     4  .36 

A      o*r 

sin  z 
m' 

sin  z 
sin  (z'  —  z) 

sin  z  sin  z' 

a  —  a'  =  18  22  .II 

«  =  2  35   39  .« 

a   =  2  17    17  .00 

Aberration  =  o  .32 

Azimuth  of  mark  =  2°  17'  i7"-32 


CHAPTER  X. 

PRECESSION.— NUTATION.— ABERRATION.— PROPER  MOTION. 

321.  The  heavenly  bodies  which  are  employed  for  any  of 
the  purposes  treated  of  in  the  foregoing  pages  are,  first,  the 
sun,  moon,  and  planets;  and.  second,  the  fixed  stars. 

In  solving  the  problems  of  practical  astronomy,  we  have 
in  most  cases  supposed  the  position  of  the  object  observed 
to  be  accurately  known.  The  co-ordinates  which  we  have 
in  most  cases  employed  are  the  right  ascension  and  declina- 
tion. 

The  motions  of  the  sun,  moon,  and  planets  are  of  a  com- 
plicated character,  and  the  prediction  of  their  places  for  any 
given  instant  belongs  to  another  department  of  astronomy. 
When  their  co-ordinates  are  required  for  any  of  the  fore- 
going purposes  they  will  simply  be  taken  from  the  American 
Ephemeris  or  a  similar  publication. 

With  the  fixed  stars  the  case  is  different ;  their  relative 
positions  change  very  slightly  from  age  to  age.  In  most 
cases  no  change  at  all  has  been  discovered. 

O 

The  apparent  co-ordinates  of  all  stars,  however,  are  vary- 
ing slowly  but  continuously,  owing  to  two  causes  which  are 
independent  of  the  star's  motion,  viz.:  first,  a  shifting  of  the 
planes  of  reference,  giving  rise  to  precession  and  nutation ; 
and  second,  an  apparent  motion  of  the  star,  due  to  the  earth's 
motion  combined  with  the  progressive  motion  of  light,  called 
aberration. 


322.  SECULAR  AND  PERIODIC  CHANGES.  559 


Secular  and  Periodic  Changes. 

322.  The  small  changes  to  which  many  of  the  quantities 
employed  in  astronomical  operations  are  subject  are  divided 
into  two  classes,  viz.,  secular  and  periodic. 

Secular  changes  are  those  which  are  progressive  in  the  same 
direction  from  year  to  year,  requiring  long  periods  of  time — 
sccnla — to  complete  a  cycle,  so  that  during  short  periods  the 
changes  may  'be  considered  as  proportional  to  the  time. 

Periodic  changes  are  those  which  complete  their  cycle  in  a 
comparatively  short  time,  and  where  the  motion  from  maxi- 
mum to  minimum,  or  the  reverse,  is  so  rapid  that  the  change 
cannot  be  considered  proportional  to  the  time,  except  for 
very  short  intervals. 

T\&  precession  of  the  equinoxes  produces  a  secular  change  in 
the  co-ordinates  of  all  stars  referred  either  to  the  equator  or 
ecliptic.  It  will  be  remembered  that  this  is  the  name  given 
to  the  slow  motion  which  takes  place  in  the  line  of  intersec- 
tion of  the  ecliptic  and  equator,  causing  the  pole  of  the  equa- 
tor to  describe  a  circle  about  the  pole  of  the  ecliptic  in  a 
period  of  about  25,000  years.  This  motion  is  due  to  the 
spheroidal  form  of  the  earth,  in  consequence  of  which  one 
component  of  the  attractive  force  of  the  sun  and  moon  tends 
to  draw  the  equator  into  coincidence  with  the  ecliptic. 
This  component  of  the  attraction  is  not  uniform.  It  is*  a 
maximum  when  the  sun  and  moon  are  farthest  from  the 
plane  of  the  equator,  and  a  minimum  when  they  are  in  the 
equator. 

Nutation.  The  want  of  uniformity  in  the  forces  producing 
precession  gives  rise  to  small  changes  of  short  period  which 
together  are  called  nutation.  There  are  a  number  of  small 
changes  embraced  under  "this  head,  but  the  principal  one 
causes  the  actual  pole  of  the  earth's  equator  to  describe  a 


560  PRACTICAL  ASTRONOMY.  §  324. 

small  ellipse  about  the  mean  pole ;  the  major  axis  of  this 
ellipse  is  directed  to  the  pole  of  the  ecliptic  and  embraces 
about  1 8"  of  arc.  The  length  of  the  conjugate  axis  is  about 
14".  The  period  is  about  18  years. 


Mean,  Apparent,  and  True  Place  of  a  Star. 

323.  Suppose  the  right  ascension  and  declination  of  a  star 
to  be  accurately  observed  with  a  suitable  instrument :  the 
place  of  the  star  so  determined  will  be  the  apparent  place. 

The  apparent  direction  of  the  star  is  affected  by  aberration, 
the  effect  of  which  will  be  considered  more  fully  hereafter. 
If  we  apply  to  the  apparent  right  ascension  and  declination 
the  corrections  necessary  to  free  them  from  the  effect  of 
aberration,  we  have  the  *true  place. 

If  now  we  apply  to  this  true  place  the  small  periodic  cor- 
rections called  nutation,  we  have  as  the  result  the  mean  place. 

In  catalogues  of  stars  the  right  ascensions  and  declinations 
are  given,  referred  to  the  mean  equator  and  equinox  for  the 
beginning  of  the  year  of  the  catalogue.  If  then  the  apparent 
place  of  the  star  is  required  for  any  given  date,  the  preces- 
sion must  be  applied  to  reduce  the  mean  place  of  the  cata- 
logue to  the  mean  place  at  the  given  date;  the  nutation  and 
aberration  must  then  be  applied  to  reduce  the  mean  place 
to  apparent  place.  The  determination  of  these  reductions 
will  be  the  immediate  object  of  the  present  chapter. 


Precession. 

324.  The  change  in  the  position  of  the  equinoxes  is  due 
to  two  causes:  first,  the  action  of  the  sun  and  moon;  and 
second,  that  of  the  planets.  The  first  gives  rise  to  luni-solar 
precession,  and  the  second  to  planetary  precession. 


§  325.  PRECESSION.  561 

By  the  processes  of  physical  astronomy  it  is  shown  that  the 
attractions  of  the  sun  and  moon  upon  the  matter  accumulated 
about  the  earth's  equator,  which  gives  it  its  spheroidal  form, 
produce  a  slow  retrograde  motion  in  the  line  of  intersection 
of  the  equator  and  ecliptic,  without  changing  the  angle  be- 
tween these  planes.  As  the  celestial  longitudes  are  measured 
from  this  line,  or  rather  from  one  of  the  points  where  it 
pierces  the  celestial  sphere,  the  effect  is  a  constant  increase 
in  the  longitudes,  with  no  change  in  the  latitudes. 

This  is  luni-solar  precession,  and  is  due  simply  to  a  motion 
of  the  equator. 

The  attractions  exerted  upon  the  earth  by  the  other  planets 
of  the  solar  system  tend  to  change  the  plane  in  which  it  re- 
volves about  the  sun,  without  changing  the  position  of  the 
equator;  this  change  is  relatively  small  and  tends  to  diminish 
the  right  ascensions  without  affecting  the  declinations. 

The  latter  is  called  planetary  precession  and  is  due  to  a  mo- 
tion of  the  ecliptic. 

The  combined  effect  of  the  luni-solar  and  planetary  pre- 
cession is  to  produce  small  secular  changes  in  the  right  ascen- 
sions and  declinations,  also  of  the  longitudes  and  latitudes  of 
all  stars,  and  in  the  obliquity  of  the  ecliptic. 

325.  In  order  to  be  able  to  determine  the  position  of  the 
equator  or  the  ecliptic  at  any  given  instant  it  will  be  neces- 
sary to  select  the  positions  of  those  circles  at  some  given 
epoch  as  fixed  circles  to  which  all  motions  may  be  referred. 
Let  these  fundamental  circles  be  the  mean  equator  and  eclip- 
tic for  1800.0. 

In  Fig.  67,  let  AA0  be  the  mean  equator  for  1800.0; 
A'A",  the  mean  equator  for  1800  +  /. 

Let  EE0  and  EE'  be  the  mean  ecliptic  for  1800.0  and 
1800  -f-  t  respectively. 

Then  BD,  the  part  of  the  fixed  ecliptic   over  which  the 


562  PRACTICAL  ASTRONOMY.  §  325. 

point  of  intersection  has  moved,  is  the  lum-solar  preces- 
sion in  t  years  =  i/>. 

Let  D'  be  the  point  on  the  movable  ecliptic  which  coin- 
cided with  D  when  the  ecliptic  had  the  position  EEQ. 

Then  CD'  is  the  general  precession  for  t  years  =  ^,. 

Since  B  is  the  point  of  the  equator  which  at  the  instant 
1800.0  was  at  D,  BC  is  the  arc  of  the  equator  over  which 


FIG.  67. 

the  intersection  with  the  ecliptic  has  moved  in  a  forward 
direction. 

BC  is  therefore  the  planetary  precession  in  the  interval 
t  years  —  3. 

Let  GOQ  =  the   mean    obliquity   of  the   ecliptic   for  1800.0 

=  A.DE-, 
GO,  =  the  obliquity  of  the  fixed  ecliptic  for  1800  +  * 

=  A"BE\ 
co  =  the  mean  obliquity  of  the  movable  ecliptic  for 

1800  +  *  =  A"CE\ 

n  =  the  inclination  of  the  mean  ecliptic  for  1800  -f-  / 
to  the  fixed  ecliptic  —  BEC. 

D  is  the  mean  equinox  of  1800;  C  is  the  mean  equinox  of 
1800  +  *. 


§  326.  PRECESSION  CONSTANTS.  563 

Since  longitudes  are  reckoned  in  the  direction  DE^  E  will 
be  the  descending-  node  of  the  movable  on  the  fixed  ecliptic. 

Let  II  —  the  longitude  of  the  ascending  node  of  the  mov- 
able on  the  fixed  ecliptic,  reckoned  from  the 
mean  equinox  of  1800. 
Then  H  =  180°  -  DE. 

326.  The  determination  of  the  values  of  the  above  con- 
stants, by  means  of  which  the  position  of  the  mean  ecliptic 
and  equator  at  any  time  1800  +  *  can  be  determined  in 
reference  to  the  fixed  ecliptic  and  equator  of  1800.0,  belongs 
to  the  department  of  physical  astronomy.  Three  different 
series  of  values  have  been  quite  extensively  employed,  viz., 
those  of  Bessel,  Struve  and  Peters,  and  Leverrier.  Bessel's 
values  are  given  for  the  mean  ecliptic  and  equinox  of  1750, 
those  of  Struve  and  Peters  for  1800,  and  Leverrier's  for  1850.0. 
The  values  which  we  shall  employ  are  those  of  Struve  and 
Peters,  being  those  which  are  more  extensively  used  at 
present  than-either  of  the  others.  If,  however,  it  is  preferred 
to  use  other  values,  it  will  be  a  simple  matter  to  make  the 
necessary  changes  in  the  formulas  which  will  be  derived. 
The  values  are  as  follows:* 

tp  =  50".  3798^  —  o.opo  io84/2; 
fa  ~  50".  24  1  it  +  o.ooo  1 


<»0  =  23°  27'  54".22; 

®,     =     <»0+    -°00  00735* 

c»=  &90  —  / 
n=  i2° 


-     .ooo  003  5/; 
—  o".i5ii9*—  .000  241  86*2. 


•    •    (523) 


*  Dr.  C.  A.  F.  Peters'  Numerus  Constans  Nutationis,  p.  66  et  71. 

f  In  the  American  Ephemeris  the  value  of  the  annual  diminution  employed 
is  o".4645,  instead  of  /;  .4738.  The  difference  is  so  small  as  to  be  practically 
almost  inappreciable. 


564  PRACTICAL   ASTRONOMY.  §  327. 

Bessel  gives  the  following  values  for  the  epoch  1750:* 


-     ".000  12 17945 /2; 
fa  =  50  .21129*      +      -000  I22I483/2  ; 
fc^  =  23°  28'  i8".o  +      -ooo  00984233*'  ; 
<tf  =  23    28   1 8  .o  —      .48368*  —  .00000272295*'; 
n  =  171°  36'  10"    -  5".2i*; 
TT  =  o7/.48892/  .000  0030719*'; 

3-  —  o  .  1 7926*        —  .    .000  2660394*'. 


^(524) 


The  folio  wing  are  Leverrier's  values,  the  epoch  being  1850 : 


^  (525) 


—  ".oooioSSi*2 ; 

^  —  50  .23465*        -f  .000  H288*2  ; 

GO,  —  23°  27'  3i".83  +  .000  00719^  ; 

GO  =  23    27  31  .83  —  -47593^  —  "-OOOOOI49/2; 
n=  173°  o'  12"        -  8/</.694/; 

7t  =  o//.4795<D/          —  .000  003 1 2  f  ; 

3"  =  o  .14672*          —  .00024174^. 


Assuming  the  values  of  the  above  quantities  to  be  known, 
we  may  now  solve  the  following  problems. 

327.  Problem  First.  To  find  the  precession  in  longitude 
and  latitude  for  any  star  between  1800.0  and  1800  +  t. 

Let  the  star  be  referred  to  a  system  of  rectangular  axes, 
the  fixed  ecliptic  for  1800  being  the  plane  of  XY,  the  positive 
axis  of  X  being  directed  to  the  ascending  node  of  the  ecliptic 
of  1800  +  t  on  the  fixed  ecliptic,  the  positive  axis  of  Z  being 
directed  to  the  pole  of  the  fixed  ecliptic. 

Let  L  and  B  —  the  longitude  and  latitude  for  1800.     Then 

x  —  cos  B  cos  (L  —  II) ;   y  =  cos  B  sin  (L  —  77) ;    z  —  sin  B.(a) 
Next,  let  the  plane  of  JFFbe  the  mean  ecliptic  of  1800  +  t, 

*  Tabulae  Regiomontanae,  p.  v,  Introduction. 


§  327.  PRECESSION.  565 

the  new  axis  of  X  coinciding  with  the  old,  and  the  new  axis 
of  Z  directed  to  the  pole  of  the  ecliptic  of  1800  -f-  /. 

Let  A  and  ft  =  the  longitude  and  latitude  for  1800  +  /. 
Then 


II  is  the  same  in  both  (a)  and  (d),  being  the  value  for  1800.0. 

The  new  axes  of  Y  and  Z  make  the  angle  n  with  the  old. 
Therefore 

x'—x\      y'=y  cos  7t  -\-  z  sin  n\      z'=—y  sin  n  -\-  z  cos  n.  (c) 
From  (a),  (b),  and  (c), 


(d)  cos  ft  cos  (A  -  n  -  ^)  =  cos  B  cos  (L  —  77); 

(i)  cos  fi  sin  (/I  —  77  —  ^0  =  cos  ^  sin  (Z  —  II)  cos  ?r  -[-  sin  B  sin  ?r;    [•  (526) 

(f)  sin  /?  =  —  cos  B  sin  (L  —  II)  sin  TT  -|-  sin 


n^sin  7T;     V 
>in  ^  cos  n.   ) 


These  equations  are  rigorous,  but  in  practice  they  may  be 
much  abridged. 

n  is  so  small  that  no  appreciable  error  will  be  involved  in 
writing  cos  n  =  i,  even  when  the  interval  /  is  several  cen- 
turies. 

Making  cos  TT  =  i,  and  multiplying  (d)  by  sin  (L  —  H), 
(e)  by  cos  (L  —  7T),  and  subtracting,  we  have 

cos  fi  sin  (A.  —  L  —  ^,)  =  sin  n  sin  B  cos  (L  —  II). 

Then  multiplying  by  cos  (L  —  77)  and  sin  (Z  —  77),  and  add- 
ing, we  find 

cos  ft  cos  (A  —  L  —  ^j)  =  cos  -5  -{-  sin  ?r  sin  B  sin  (Z'  —  71); 
and  by  division, 

.  sin  n  tan  B  cos  (Z  —  77) 

tan  ; 


566  PRACTICAL  ASTRONOMY.  §  327. 

Developing  this  into  a  series  and  writing  sin   n  —   TT,  we 
have* 

A  —  L  —  $1  =  n  tan  B  cos  (L  —  II)  —  ^  tan8  B  sin  z(L  —  II)  —  etc.,  (527) 

where  the.  term  in  n*  may  always  be  omitted. 
The  last  of  (526)  may  be  written 

sin  ft  =  sin  B  —  sin  n  cos  B  sin  (L—  II). 

/?  is  a  function  of  n.     Developing  by  Maclaurin's  formula, 
we  have 

/?  —  B  =  —  n  sin  (L  —  H)  +  %n*  tan  B  sin2  (L  —  77),  etc.  (528) 

Formulae  (527)  and  (528)  solve  the  problem,  where,  as  be- 
fore remarked,  the  terms  in  7r2  may  always  be  dropped. 

*  This  expansion,  which  is  of  frequent  application,  is  obtained  as  follows: 
Writing  (A  —  L  —  ^)  =  x,  it  tan  B  =  m, 


m  sin  y.          sn  x 

the  above  formula  becomes  tan  x  —  —  -  —     -  =  —  —  . 

I  -f-  m  cos  y       cos  x 

From  this  we  have"  sin  x  =  m  sin  (y  —  x).  , 

Adding  both  members  to  m  sin  x,  then  subtracting  both  members  from  m  sin  x 
and  dividing, 

m  -f-  i  _  sin  x  -j-  sin  (y  —  x}  _         tan  \y 


m  —  r  ~~  sin  x  —  sin  (y  —  x)       tan  (x  —  iy) 

N9W  write          ^  ~  '  =/;        .*  -  |j  =  w;        |^  =  ».          tan  w  =/  tan  v\ 
and  by  Moivre's  formula,  equation  (135), 

2u  V^l  2u  V^l 


§  328.  PRECESSION.  $6? 

328.  Problem  Second.     To  find  the  precession  in  longitude 
and  latitude  between  two  given  dates  1800  +  /  and  1800  +  /'. 

Let  A  and  ft  be  the  longitude  and  latitude  for  1800  -f-  /; 
A/  and  ft'  be  the  longitude  and  latitude  for  1800  +  '/'. 

Then  by  (527),  A   .-.£==£   +  n  tan  B  cos  (L  -  H  ); 
\'  -  L  =  t/,\'  +  n'  tan  B  cos  (L  -  II'). 

Subtracting, 


This  may  be  placed  in  a  better  form   by  assuming  the 
auxiliary  equations 


a  sin  A  =  (*'  +  T)  sin  i(77'  -  ^);  I 
a  cos  A  =  (n1  -  -  a)  cos  J(/T  -  77).  I   ' 


2V  V  _ 
_  .  .  -      ,  ~=~  -         f~=  I 

From  this  we  find       e 


2  (u  +  v)  V  —  l          1.  -\-  me 


I  -j-  me 


Taking  the  logarithms  of  both  members  of  the  above  and  expanding, 

I  A/ ~ _  2"  V'~^~l  _     1      2      *r  ^^     I      1      3      6u  ^^-^ 

Or  u  -\-  v  =  m  sin  2v  —  \ni*  sin  \v  -\-  \m*  sin  6v,  etc. . 

Writing  for  «,  v,  and  w  their  values,  we  have 

A  —  z  _  1pl  =  n  tan  B  cos  (Z  —  77)  —  \-rP  tan2  ^  sin  z(L  —  77) 

—  ^7r3  tan3  B  sin  3(Z  —  77),  etc. 


5^8  PRACTICAL  ASTRONOMY.  §  328. 

Combining  these  with  (529),  and  eliminating  n  and  TT',  we 
find 


_  \  =  fa'  -  ^)  _[.  a  cos  \L  --  i^7  - 


.  (531) 


Similarly  from  (528)  we  have  for  1800  +  t  and  1800  -f  t' 

ft   -  B  =  -  n  sin  (Z  -  77  ); 
fl'>—  £  =  -  ar'sin(£  -  /I7). 

Subtracting  and  eliminating  n  and  TT'  by  the  auxiliary  equa- 
tions (530),  we  find 

ft'  -  ft  =  „  a  sin  (L  -  ~~—    -  A).    .     (532) 
For  the  auxiliary  quantities  a  and  A  we  find,  from  ($30), 
tan  ^  =        ~  tan  |  (ZI7  -  77). 


If  we  substitute  for  TT  and  TT'  their  values  from  (523),  neg- 
lecting the  term  in  f,  and  recollecting  that  i(7I'  —  71)  is  very 
small,  this  equation  may  be  written 


/ 

.    (533) 


being    therefore  very   small    even   for  large   values   of 
and  /',  we  may  write  cos  A  —  i  in  (530),  when 

a=n'  -n  =  (t'  -  f)  ".4776  -  (*"  -  f)  ".0000035.  (534) 


§  329.  PRECESSION.  569 

In  equations  (531)  and  (532)  we  may  write  A  —  ^x  for  Z, 
and  fi  for  B.     Introducing  the  auxiliary  angle  M  such  that 


L  - 


and  substituting  in  (531),  (532)>  and  (534)  for  #/,  #'  ?r,  77, 
Struve  and  Peters'  values — equation  (523) — we  have  finally 
the  following  practical  formulae  for  computing  the  preces- 
sion in  longitude  and  latitude  between  any  two  intervals 
1800  4-  /and  1800  +  /': 

M  =  172°  45'  3i"  +  /  5o".24i  -  (/'  -f  t)  8". 505; 
A'-A.=       (/'  —  /)  [50". 241 1  +  (/'  +  /)  o".ooo  1134]  M536) 

-f  (f  —  /)  [  o".4776  —  (/'  +  /)  o".ooo  0035]  cos  (A.  —M)  tan  /?; 
/$'  —  ft  =  —  (?  —  *)[  o".4776  —  (t'  -j-  /)  o".ooo  0035]  sin  (A.  —  M). 

329.  If  we  divide  the  expressions  for  (A/  —  A)  and  (/?'  —  fi) 
by  (t'  —  /),  and  then  make  /  =  t',  we  shall  have  the  values 

of  —j-  and  — j-j  or  the  expressions  for  the  precession  in  longi- 
tude and  latitude  respectively  at  the  instant  /,  viz. : 


-jr  —       5o/x.24i  i  4~  o.ooo  2268/; 

+  [o//4776  —  o.ooo  0070/1  cos  (\  —  M}  tan  /?; 

-//? 

-77  —  —  [o"-4776  —  o.ooo  0070?]  sin  (X  —  M). 


-(537) 


These  formulas  may  be  used  to  compute  the  entire  pre- 
cession between  two  dates  1800  +  /  and  1800  -f-  /',  if  we 
compute  the  values  of  the  differential  coefficients  for  the 
middle  interval,  viz.,  1800  +  %(t  +  /').  .  The  result  will  be 
accurate  to  terms  of  the  second  order  inclusive. 


5/0  PRACTICAL  ASTRONOMY.  §  329. 

We  have  developed  these  formulae  (536)  and  (537)  (which 
are  those  of  Bessel,  except  that  we  have  employed  other 
constants)  for  the  sake  of  completeness,  although  they  will 
not  be  used  in  connection  with  the  problems  of  the  present 
treatise,  the  co-ordinates  commonly  employed  being  the 
right  ascension  and  declination. 

Example.  The  mean  longitude  and  latitude  of  a.  Lyra  for  1850.0  are  as  fol- 
lows: 

A  =s  283°  12' 48".  12; 
ft  =    61    44  25    .45. 

Required  the  mean  longitude  and  latitude  for  1884.0. 

Here  t  =  50;        f  =  84;        /'  —  /  =  34;        t1  -f  t  =  134. 

Therefore  we  find,  by  (536), 

M=  173°  8'  23"; 
A  —  M=  no   4   25; 

X  -  A  ="  (f  -  t}  X  so".256'3  +  (f  -  t}  X  .4771  cos  (A  —  M)  tan  ft; 
ftT-ft=-(f-t)X  .4771  sin  (A  -  JIf ). 

A'  -  A  =          28'  i8".36  ft'  -  ft  =  -          15". 24 

A  =  283°  12'  48".i2  ft  =  61°  44'  25". 45 

A'  =  283°  41'    6".48  ftf  =  6i°44'  io".2i 

If  we  wish  to  employ  (537),  we  shall  have  for  t  the  middle  of  the  interval  be- 
tween 1850  and  1884,  viz.,  t  =  67.  For  A  in  the  second  member  we  require 
the  longitude  for  1867,  which  we  shall  have  with  all  necessary  accuracy  by 
adding  to  the  longitude  for  1850  the  general  precession  for  17  years  and  neg- 
lecting the  smaller  terms.  Calling  this  value  A0,  we  have 

Ao  =  283°  12'  48"  +  50". 24    X  17  =  283°  27'    2"; 
M=  172    45  3i     +33   -231  X  67  =  173    22   37; 
AO  —  M  —  no     4   25; 

~  =  50". 2563  -f  .4771  cos  (A0  -  M)  tan  ft  =  49".95i7; 

d& 

--=  ~  -4771  sin  (A0  -  M)  =  -".4481. 


§  330.  PRECESSION.  5/1 

Therefore 


PRECESSION. 
A'  -  \  -      (i>  -  t)  =  28'  i8".36j 


agreeing  with  the  values  obtained  by  the  other  formulae. 

330.  Problem  Third.  Given  the  mean  right  ascension  and 
declination  of  a  star  for  the  date  1800  +  t,  required  the  right 
ascension  and  declination  for  1800  +  /', 

We  first  require  the  values  of  certain  auxiliary  constants 
similar  to  those  employed  in  solving  the  corresponding  prob- 
lem for  the  ecliptic.  - 


FIG.  68. 


In  Fig.  68  let  Vy{  —  the  fixed  ecliptic  for  1800; 
^  —  the  equator  for  i8oo  +  /; 
S  =  the  equator  for  1800  +  /'; 
Vl  V{  =  the  luni-solar  precession  in  the  in 
terval  (tf  —  t). 

Therefore  V^'  =  $'  -  ^. 

Let  QV,  =  90°-^;    QV;=cp°+z';     V&V^V. 

z,  z'  ,  and  6  will  be  quite  small  quantities,  even  when  the  in- 
terval (/  —  /)  is  considerable. 

In  accordance  with  our  notation,  angle  gFjF/—  180°—  <»„ 


Then  in  the   triangle   QVyf  the  quantities  o?/, 


572 


PRACTICAL  ASTRONOMY. 


§331 


$'  —  $  are  given  by  (523);  we  can  therefore  determine  z,  z' , 
and  (9. 

By  Napier's  analogies,  we  readily  find 


The  second  of  these  may  be  written 


(538) 


In  the  first  and  third  the  denominator  may  be  written  equal 
to  unity. 

331.  We  can  now  solve  our  problem,  viz.,  to  determine 
the  right  ascension  and  declination  for  1800  +  /',  having 
given  those  quantities  for  1800  -f-  /. 

In  Fig.  68,  5  being  any  star,     Sa  =  #,     Sa'  =  tf'. 

If  F3  and  F/  represent  the  position  of  the  mean  equinox 
for  1800  +  /  and  1800  +  1'  respectively,  then 

The  planetary  precession  in  the  interval  /  —  J7  J^  —  5; 
The  planetary  precession  in  the  interval  t'  =  F/F/=  5r. 


The  right  ascension 


=  a; 
=  a'- 


V^Q  —  90°  —  z  —  5; 
VJQ  =  90°  +  z'  -  $' 


Considering  now  the  rectangular  co-ordinates  of  the  star, 


§  331-  PRECESSION.  573 

the  mean  equator  of  1800  -f-  1  being  the  plane  of  XY,  the  posi- 
tive axis  of  X  being  directed  to  the  point  Q,  we  have 

x  =  cos  d  sin  (a  -f-  z  -|-  $); 
y  =  COS  d  COS  (tf  +  *  +  ^); 

#  =  sin  #. 
Similarly  for  the  equator  of  1800  +  /', 


/  =  cos  *'  cos  (af  -z'  +  S')J 
*'  =  sin  <T. 

The  formulae  for  x',  y',  and  *',  in  terms  of  x,  y,  and  z,  are 

AT7  =  X', 

y'  =  y  cos  8  —  z  sin  #; 
#'  =  y  sin  0  +  *  cos  6. 

Therefore 

cos  6'  sin  (a1  -*'-{-  5')  =  cos  S  sin  (a  4-  z  -f  5);  j 

cos  d'  cos  (a'  —  2'  +  3')  =  cos  d  cos  (a  +  z  -f  5)  cos  6  —  sin  5  sin  0;  j-  .  (539) 
sin  d'  =  cos  &  COS  (a  -j-  *  -f  3)  sin  6  -f  sin  d  cos  0.  ) 

We  might  have  derived  these  equations  by  applying  the 
formulae  of  spherical  trigonometry  to  the  triangle  P/ 
formed  by  joining  the  place  of  the  star  with  the 
pole  of  the  equator  in  the  two  positions. 

Thus  in  Fig.  69,  5  being  the  star,  and  P  and  P  the 
pole  of  the  equator  at  the  time  i8oo-f*  and  1800+*' 
respectively,  we  have  the  following  for  the  sides 
and  angles  of  the  triangle.  Calling  the  angle  at 
the  star  C,  FIG.  ^ 

PP  =  6;  PS=  90°  -tf;        /*S=90°-<T; 

SPP  =  a-\-z-\-  $  =  A,  say,  for  convenience; 

SPP  =  i8o°—a'-*'S'  =  180°  -A'. 


574 


PRACTICAL   ASTRONOMY. 


§  332. 


Another  solution  of  the  problem  is  obtained  by  applying 
Gauss'  equations  to  this  triangle,  viz.: 


cos 


cos  4(90°+*')  sin  $(A'-+C)=cos  i(9o°+(?-#)  sin  %A\ 
sin  4(90°+$')  cos4(-4'-Q=sm  4(90°  +3+0)  cos  %A\ 
sin  490°+*  sin  4^—  =sin  £o0tf-6>  sin  A 


(540) 


The  auxiliary  quantities  zy  z't  and  6  being  computed  by 
(538),  either  (539)  or  (540)  give  the  required  solution  of  our 
problem;  these  equations  being  solved  in  the  usual  manner. 

332.  Practically  it  is  more  convenient  to  compute  the  dif- 
ferences, (af  —  a)  and  (#'  —  d).  A  formula  for  (a1  —  a)  is 
conveniently  derived  from  the  first  and  second  6f  (539), 
which  we  write  as  follows: 

cos  $'  sin  A'  =  cos  #  sin  A', 

cos  <$'  cos  A  '  =  cos  <$  cos  A  cos  6  —  sin  <5  sin  6. 

Multiply  the  first  of  these  by  cos  A,  the  second  by  sin  A,  and 
subtract;  then  multiply  the  first  by  sin  A,  the  second  by 
cos  A,  and  add.  We  readily  find 


cos  8'  sin  (A1  —  A)  —  cos  8  sin  A  sin  0  [tan  8  -f  cos  A  tan  $0]; 
cos  5'  cos  04'  —  A)  —  cos  5  —  cos  8  cos  ^  sin  0[tan  8  -f-  cos  A  tan  $0] 

Let  /  =  sin  ^[tan  8  +  cos  A  tan  40], 


)      /       , 
,    f 


I— 


By  the  first  of  Napier's  analogies, 
tan  KiT  -  «J)'= 


. 

cos  \(A  —A) 


B  ^  (542) 


§332.  PRECESSION.  ,575 

It  will  be  necessary  to  make  the  computation  in  this  com- 
plete form  for  circumpoiar  stars  when  the  interval  (/'  --  /)  is 
large.  When  the  star  is  not  too  near  the  pole  the  computa- 
tion will  be  much  simpler,  as  we  shall  see. 

Example.     The  mean  jflace  of  Polaris  for  1825.0  is  as  follows: 

Right  ascension  a.  —    ou  58™  15'. 32; 

=  14°  33'  49"-8. 
Declination  d  =  88°  22'  3*".47. 

Required  the  precession  in  right  ascension  and  declination  between  1825  and 
1900. 

We  have  here  /  —  25,  t'  =•  100.     We  therefore  find,  from  formulae  (523), 

<»!  =  23°  27'  54". 22459;         ^  =  i259"-43;          5  =    3"-628; 
K>!f  =  23    27   54  .29350;        ^  =  5036  .90;        $'  =  12   .700. 

' 
Then  by  formulae  (538),  which  we  may  write 

tan  *(*'  +  z)  =  cos  l(ftV  +  coO  tan  |(^'  -  ^), 

i(s'  -  *)  =  KCO/  -  ooO  cot  i(^'  -  ^)  cosec  K^i'  +  »i), 

tan  £0  =  sin  \(z'  -f  2)  tan  i((»/  +  o^). 

|(^  _  ^,)  —    31' 28". 74    tan  =  7.9617592     cot  =  2.03824  . 
-f  a?/)  =  23°  27'  54". 26    cos  =  9.9625128    cosec  =  .39991 
zf  -\-  z)  =  o°  28  52  .55    tan  =  7.9242720 


—  <»,)  =  0^.03446  log  =  8.53732 


9.45  -       log  -HZ'  -  4  =  0.97547 


2'  =  o°  29'    2".oo  tan  i(o>j  +  <»»')'=  9-6375775 

z  =  o    28   43   .10  sin  $(s'  -j-  z)  =  7.9242567 


tan  -J0  =  7.5618342 
£6  =o°i2'32".o7 
0  =o  25    4  ,14 


576  PRACTICAL  ASTRONOMY.  §  333. 

We  now  compute  (a'  —  a)  and  (6"  —  S)  by  formulae  (542),  viz. : 

<*  =  14°  33'  49"-8  tan  $0  =  7.56183 
2  =  28  43  .10  cos  A  =  9.98486 
S  =  3  -63 

Sum  =  7.54669 
A  =  15°  2' 36". 53  Zech  =  .434 

tan  5  =  1.5472620 

sin  0  =  7.8628593 


log/  =  9.4101647 
sin  A  =  g.  4142243 
log/  =  9.4101647 
cos  A  =  9.9848553    i(A'—A)  =  2°  32'  13". 06   sec  =  0.0004259 


/  cos  A  =  9.3950200 
Zech  =    .1239697 

log  denominator  =  9.8760303 
log  numerator  =  8.8243890 


i(A'-\-A)  =  17    34  49  .60      cos  =  9.9792268 
tan|0  =  7  5618342 


>'— d)  =  o  ii  57  .65   tan  =  7.5414869 
-  d  =  o  23  55  .30 


tan  (A1  —  A)  =  8.9483587 

A'  —  A  =    5°    4'  26".I3 
A  =  15      2  36  .53 


A'  =  20°    7'    2". 66 

'-A)  =    5°    4'26".I3 
*)  =          57  45  .10 
_  ($>  -  3)  =  -  9  .07 

a'  —  a  =    6°    2'    2".i6 
=    oh  24m  8".  144 

333.  By  means  of  the  foregoing  formulae  we  readily  find 
the  precession  in  right  ascension  and  declination,  viz.,  -^ 

dd 
and  -T-,  at  any  given  instant  1800  +  /. 

We  have  (A1  -  A)  =  (a'  -a)-  (z1  +  *)  +  (V  -  5).   (543) 


§  333-  PRECESSION.  5/7 

If  now  we  make  /'  =  /  in  the  first  of  (541),  we  may  make 
6'  ±=  tf,  sin  (A'  -  A]  =  A'  -  A,  sin  6  =  6,  sin  A  =  sin  (or+S); 
also,  sin  6  tan  £0  will  vanish,  being  an  infinitesimal  of  the 
second  order. 

Therefore  this  equation  becomes 


+  $).     .     .     .     (544) 
From  (538),  the  same  condition  existing,  viz.,  t  —  t  ',  we  have 


=   #  -  *  cos 
=      '-      sin 


Combining  (543),  (544)  and  (545),  writing  da,  d$,  and 
for  (a'  —  a),  etc.,  and  dividing  by  dt, 


da  d$      dfy  .   dfi    .  *    • 

=  -  -  cos  ^  sm  ^   tan  d  sm 


The  last  of  (542)  by  a  similar  process  gives 

^  =  ^8111^  cos  (a +  3) (547) 


Writing  m  =  —  -jr  +  -jr  cos  &„ 

^  Y *(548) 

*  =  W sin  °''- 


*  If  we  draw  in  the  plane  of  the  equator  lines  to  the  mean  equinox  o 
and  (1800  -f-  t  -j-  i)  years,  it  will  be  observed  that  m  represents  the  angle  be- 
tween them,  assuming  the  rate  of  change  to  be  uniform  during  one  year.  Also, 
n  will  be  the  angle  between  the  two  lines  drawn  to  the  poles  of  the  equator  in 
the  two  positions. 


$,-8  PRACTICAL  ASTRONOMY.  §  334. 

From  the  values  of  4\  <*>!>  an^  ^ — equation  (523)— we  have 


m  =  46".  0623  +  ".ooo  2849/5 
=   3s.o/o82  +  s.ooo  01899/5 
n  —  2o//.o6o7  -   '".ooo  0863/5 

—  —  m  >-{-  n  sin  <*  tan  6; 


-— 
at 


—  n  cos 


(549) 


We  have  written  a  in  place  of  (or -|- -5),  no  appreciable 
error  resulting  from  neglecting  3-. 

These  formulae  may  be  employed  for  computing  the  pre- 
cession between  any  two  dates  1800  +  /  and  1800  -f  /'.  If  the 

J  i   OL 

values  of  —r-  and  —=-  are  computed  for  the  middle  date,  viz., 

1800  +  \(t  -\-  /'),  the  result  will  be  accurate  to  terms  o(  the 
second  order  in  ••(/-'  —  t)  inclusive.  We  shall  return  to  these 
formulae  hereafter. 

Proper  Motion. 

334.  When  the  co-ordinates  of  a  star  observed  at  different 
dates  are  reduced  to  the  same  epoch  by  means  of  the  pre- 
cession formulae,  a  considerable  difference  in  the  values  is 
often  found,  indicating  a  motion  of  -the  star  itself.  This 
change  is  called  proper  motion,  and  may  be  due  either  to  an 
actual  motion  of  the  star  in  space  or  to  the  motion  of  the 
solar  system,  producing  an  apparent  motion  of  the  star.  The 
observed  proper  motion  is  in  fact  the  resultant  of  the  two. 
For  our  purposes  it  is  not  necessary  to  attempt  to  separate 
these  components.  The  proper  motions  in  most  cases  are 
very  small,  requiring  many  years  to  produce  an  appreciable 
change  in  the  star's  place;  but  there  are  a  few  important  ex- 
ceptions to  this  rule. 


§  334-  PROPER  MOTION,  5/9 

In  investigating  the  subject,  the  path  of  the  star  is  assumed 
to  coincide  with  a  great  circle,  and  the  motion  to  be  uniform. 
It  is  not  probable  that  either  assumption  is  true,  but  such 
deviations  as  may  exist  will  be  very  small. 

In  order  to  determine  a  star's  proper  motion,  its  place 
must  be  'observed  on  at  least  two  dates  which  we  may  call 
1800  +  t  and  1800  +  /'.  The  greater  the  interval  (f  --  t) 
the  more  accurate  will  be  the  results,  other  things  being 
equal. 

Let  <*  and  S  =  the  observed   mean  right  as- 

cension and  declination  for 
1800  +  /; 

a  +  AOL  and  d  -\-  Ad  —  the    values    given    by    reduc- 

ing the  values  observed  at 
1800  +  t'  to  the  first  date  by 
the  application  ol  the  pre- 
cession only. 

Then  Act  and  Ad  will  be  the  changes  in  a  and  6  due  to  proper 
motion  in  the  interval  (t'  —  t). 

Let  yw  and  //  =  the  annual  proper  motion  in  right  ascen- 
sion and  declination  respectively. 

Act  AS 

Then  p  =     -  /  =     —  --  .....     (550) 


These  values  will  be  referred  to  the  mean  equator  of 
I3OO  _|_  tm  If  We  had  reduced  the  co-ordinates  for  this  date 
to'iSoo  +  *''we  should  have  obtained  the  proper  motions 
referred  to  the  equator  of  the  latter  date  : 

Act'  Ad' 

*  =  --         and         '•  =  •   •   • 


• 


580  PRACTICAL   ASTRONOMY.  §  336. 

These  values  for  stars  near  the  pole  may  differ  very  con- 
siderably from  the  first. 

335.  Problem  I.     To  reduce  the  right  ascension  and  dec- 
lination of  a  star  from  the  epoch  1800  +  /  to  1800  +  t',  the 
proper  motion  being  known. 

First.  Suppose  the  proper  motion  given  in  reference  to  the 
mean  equator  of  1800  +  /,  the  solution  is  as  follows: 

Add  to  the  right  ascension  for  1800  + 1  the  effect  of  proper 
motion  for  the  interval  (/'  —  t\  viz.,  jw(/'  —  /);  similarly  add 
to  the  declination  //(*'  •-  *)•  With  these  values  of  the  right 
ascension  and  declination  the  precession  is  computed  as 
before  by  formulae  (542). 

Second.  The  proper  motion  being  given  for  the  mean 
equator  of  1800  +  /'. 

Reduce  the  star's  place  to  1800  +  t'  by  formulas  (542),  and 
add  to  the  results  p(t'  -  -  f)  and  p'(t'  —  t)  respectively. 

336.  Problem  II.     Having  given  the  proper  motion  in  right 
ascension  and  declination,  referred  to  the  mean  equator  of 
I800  _|_  t,  to  derive  the  values  in  reference  to  the  equator  of 
1800  +  t'. 

Equations  (539),  giving  the  values  of  a'  and  &  in  terms  of 
a  and  d,  are  as  follows : 

cos  d'  sin  (a1  —  z  -f  5')  =  cos  d  sin  (a  -f-  z  -f  5);  \ 

cos  d'  cos  (a'  -  z'  +  *')  =  cos  d  cos  (a  -j-  z  +  5)  cos  0  -  sin  d  sin  0;  j-  (552) 

sin  d'  =  cos  d  cos  (a  -J-  z  -f-  3)  sin  0  -f-  sin  d  cos  0.   ) 

We  also  have 

cos  8  sin  (a  -\-  z  -f-  3)  =       cos  d'  sin  (a'  —  z'  -j-  S') ;  } 

cos  5  cos  (a  -}-  *  +  3)  =       cos8'cos(a'  —  z' -\-  $')cos  6-fsin  <5' sin  fl;  !•  (553) 

sin  d  =  —  cos  8'  cos  («'  —  z  -f-  3')  sin  0  +  sin  ^'  cos  e-   ' 

The  proper  motion  which  changes  the  position  of  the  star 
itself  produces  no  change  in  the  quantities  z,  z' ,  S",  $' ',  or  0, 
as  these  quantities  merely  serve  to  fix  the  positions  of  the 


§337-  PROPER  MOTION.  58 1 

reference  planes.  Therefore,  proper  motion  alone  being 
considered,  these  quantities  will  be  constants,  a,  a',  S,  6f 
being  variable. 

Differentiating  the  first  two  of  (552)  on  this  hypothesis, 
we  have 

cos  d'  cos  (a1  —  z  -\-  5')  da'  —  sin  d'  sin  (a1  —  z'  +  £')  dd' 

=  cos  8  cos  (a  +  z  +  3)  ata  -  sin  S  sin  (a  -f  *  +  3)  </S ; 
-  cos  d'  sin  («'  -  z'  +  3')  <ta'  -  sin  5'  cos  (a'  -  z'  +  3')  atf ' 

—  _  cos  d  sin  (a+z+3)  cos  Qda— sin  S  cos  (07+2+3)  cos  Qdd— cos  5  sin  Qdd. 

Multiply  the  first  of  these  by  cos  (af  —  z'  -f-  3'),  the  second 
by  sin  (V  —  z'  -f-  S'),  subtracting  and  reducing  by  (552)  and 
(553);  then  multiply  the  first  by  sin  (a'  --  z'  -f-  3'),  the  sec- 
ond by  cos  (a'  —  z'  +  3'),  add,  and  reduce.  We  find 


'(554) 


da,  dS,  daf,  and  dd'  have  been  changed  to  Aa,  AS,  etc. 

These  equations  solve  the  problem  above  enunciated  with 
all  necessary  precision  ;  Aa,  Ad,  etc.,  being  so  small  that  it 
is  unnecessary  to  consider  terms  of  the  higher  orders.  They 
may  be  used  'for  the  entire  proper  motion  between  the  two 
dates  t  and  /'  or  for  the  annual  proper  motion. 

337.  Problem  III.  The  proper  motion  being  given  in 
reference  to  the  mean  equator  of  1800  +  /'  to  derive  the 
values  of  Aa  and  Ad  in  reference  to  the  mean  equator  of 
1800  +  /. 

Differentiating  equations  (553)  and  reducing  by  (552)  and 
(553)  iQ  a  manner  similar  to  that  explained  above,  we  have 

Aa  =  Aa'  [cos  0  -  sin  0  tan  5  cos  (a  +  z  +  *)]  -  ~  sin  9 


Aa'  =  Aa  [cos  0  +  sin  0  tan  8'  cos  (a'  -  z'  +  d')]  + -^  sin  0  *'"  (a' i^-^; 

COS  0  COS  0 

AS 
AS'  =  -  Aa  sin  0  sin  (a'  —  z'  +  #0  H ~  cos  6'  [cos  0  +  sin  d  tan  5'  cos  (a'  — 


A3'  K5S5) 

A8  =  Aa'  sin  9  sin  (a  +  z  +  #)  +         l  cos  s  tcos  0  —  sin  0  tan  S  cos  (a  4-  z  +  #)]. 


582  PRACTICAL  ASTRONOMY.  §  337. 

Example. 

In  the  example  Art.  332  we  have  found  by  applying  the 
precession  to  the  catalogue  place  of  Polaris  the  mean  posi- 
tion for  1900.0,  as  follows: 

a'  -  Aa'  =  ih  22m  23*46;         d'  -  Ad'  =  88°  46'  26".  77. 
From  Newcomb's  catalogue  we  find  for  1900* 

a1  =  ih  22m  33s.  76;  6'  =  88°  46'  26".66. 

Therefore  Act'  =  +  ios.3o  ;  Ad'  =  —  ".n. 

t'  —  t  —  75  years.     Therefore 

/*  =  +  8-i373;        X  =  "  ."00147. 

These  values  are  referred  to  the  mean  equator  of  1900. 
If  we  wish  to  reduce  them  to  the  equator  of  1825  we  employ 
formulae  (555).  From  the  values  of  (a  -±-  z  +  3)  and  0, 
Art.  332,  we  find 


da.'  [cos  6—  sin  0  tan  d  cos  (tt-f-H-^)]  =       78-742 

48'  sin  0  sin  (or  -f  z  -4-  3) 

—  f  —    =          -023 
15  cos  o   cos  o 


=  -|-  7s-  765     Therefore  ju=-\-*.io3$ 


Also,  f  J5  ^a'  sin  6  sin  (or  +  2  +  5)  =  +".2924 

—  -r-,  cos  5  [cos  0—  sin  0  tan  d  cos  (a-f-z+3)]  =  —   .  1096 


The  above  treatment  of  the  problem  is  due  to  Bessel. 

*  This  is,  of  course,  not  an  observed  place,  but  it  answers  equally  well  for 
illustrating  the  method. 

f  Aa'  being  given  in  time  and  AS'  in  arc. 


§  339-  EXPANSION  INTO   SERIES.  583 


Proper  Motion  on  the  Arc  of  a  Great  Circle. 

338.  Let  p  =  the  annual  motion  on  the  arc  of  a  great  circle; 
X  =  the    angle    which   this    great    circle   forms 
with  the  hour-circle  of   the  star. 
When  the  star  is  on  the  meridian, 
X    will    be    measured    from    the* 
north  towards  the  east. 

In  the  figure  P  is  the  pole,  5  and  S'  the  first  and 
second  positions  of  the  star  respectively. 

SS'  =  p;        PSS'  =  K        SA  =  Ad  =  p  cos  X; 
S'A  =  4acosd  =  p  sin  X;         p*  =  Jtf*  +  Jo*  cos 


Expansion  into  Series. 

339.  The  foregoing  problem  of  reducing  the  mean  place 
of  a  star  from  one  epoch  to  another  is  treated  in  a  very  con- 
venient and  elegant  manner  by  expansion  into  series  in  terms 
of  the  time. 

If  we  let  of0  and  #0  =  the  right  ascension  and  declination 

for  any  time  T, 

a  and  tf   =  the  right  ascension  and  declination 
for  any  time  T  -f-  t, 

we  have  by  Maclaurin's  formula 

i  <t*a~ 

eta 


2 

i 
J'  +  ^    etc- 

When  precession  and  proper  motion  are  both  considered, 


r 
L 


584  PRACTICAL  ASTRONOMY.  §  339. 

the  changes  in  a  and  d  are  functions  of  these  two  independ- 
ent variables,  and  f"jjr|»  I  ~ji  L  etc«>  are  the  total  differential 

coefficients   with   respect   to   both    precession    and    proper 
motion. 

If  we  write  dpa,  dp§  to  indicate  a  variation  due  to  pre- 
cession, and  d^a,  d^d  to  indicate  changes  due  to  proper  mo- 
tion, we  fiave 


rda~}  _^a       d^         pttl  _  dp3 

\_dt\-~dt~  "dt\      VdtA    ~-~dt 


r 

V 


d*a~\  _  dp   <*    ,     ~  d»d»a    ,    d*a. 
df  J      ~^/2    "^        ^//2     "^   ^/a' 


and  similarly  for  the  other  coefficients. 
Equations  (549)  give  us  ~  and  -J-,  viz., 

dpa 

-~  =  m  -\-  n  sin  ex  tan  o; 


—  n  cos  a. 


(559) 


at 
Differentiating  these,  we  have 

^  =  —  +  —  sin  2a+  \~  sin  a  -f  mn  cos  a  "Itan  5  +  »"  sin  za  tan"  5; 
dt*         at         2  I-  at 

d*&  ,   dn 

-±—  =  _  mn  sm  a  +  ---  cos  a  —  w2  sin2  a  tan  5; 


mn*    .    3         „  ,    3         **     • 

=  --  \-  -  mn*  cos  za.  -\--  n  —  -  sin  2a 
22  z       at 


d"  •    ~ •          (560) 


^_  j~(2*a  —  w2  +  3»a  cos  2a)  n  sin  a  -f-  ^2W  -~-  +  n  ~  jcos  a  J  tan  5 
-f-  J3»i«3  cos  2a  +  3«  —  -  sin  2a    tan8  8  -(-  2«*  sin  a  (i  +  2  cos  20)  tan*  & 
=  -(««^  +  «^)sin  «-(*,»  +  »'  sin«  a)  «  cos  a 

-    ^  m»»  sin  za  +  3«        sin»  a     tan  8  -  3»»  sin»  a  cos  o  tan"  8. 


§  340-  EXPANSION  INTO   SERIES.  585 

340.  Let  us  now  consider  proper  motion. 

p,  Xy  V,  and  //  have  the  same  significance  as  before,  Articles 
334  and  338. 

a'  and  6'  =  the  right  ascension  and  declination  at  end  of 
time  /,  proper  motion  alone  being  considered. 

In  the  triangle  formed  by  the  pole  and  the  two  positions 
of  the  star  we  have  p 

PS  =  90°  -  tf;        PSf  =  90°  —  tf';        SS'  =  tp\ 
S'PS=a'-a;         S'SP=X- 

Therefore 

sin  <$' = sin  <5  cos  p^+cos  d  sin  pt  cos #;  )  sx 

cos^/cos(rtr/— <*)=:costf  cosp/— sin  dsinp/cosj;  M56i) 
cos  tf7  sin(of/— a)=  sin  p/  sin  j.  )  » 

v  '  FIG.  71. 


Also,  p  sin  *  =  ;i  cos  tf;    p  cos  ^  =  X;     Pa  =  (>9  cos2 


Differentiating  the  first  of  (561)  with  respect  to  tfr  and  /,  we 
find 


cos  $'  --  =  —  p  sin  d  sin  pt  +  cos  S  cos  pt.p  cos  ^. 


Substituting  for  p  cos  *  its  value  X,  and  making  t  =  o,  we 
have 


Differentiating  a  second  and  third  time  and  reducing  in  a 


586 


PRACTICAL   ASTRONOMY. 


§  341. 


similar  manner,   we  have  the  following  partial   differential 
coefficients  with  respect  to  /*'. 


.  (562, 


In  a  similar  manner,  by  differentiating  the  third  of  (561), 
making  t  =  o,  and  reducing,  we  find 


]  (563) 


Du.  .      ^u 

341.  For  the  terms  -  *         and       ,  ,     wre  differentiate  (559) 

with  respect  to  ^  and  //»  viz., 


Substituting  for  --  and  --  the  values  given  above,  we 


have 


p* 


cos  a  tan 


=  —  np  sin  a. 


•     (564) 


Therefore,  from  (558),  (560),  (562),  (563),  and  (564), 

-  =  m  -f-  n  sin  tan  d  -f-  p ; 

-  |  =  n  cos  a-\-  X; 


-•  .  .  (565) 


341.  EXPANSION  INTO   SERIES.  587 


i'  sin  a  +  sin  a+(»z  +  a/»)  «  cos  a  +2jaM       tan  6 

+  aw  sin  a(w  cos  a  -|-  p.')  tan2  6; 

P^l  =  _  {m  +  2lji)n  sin  a  +  •£  cos  a  -  ^  sin  28  -  »2  sm«  a  tan  5. 

Also  we  have 


+3       8 


-^?r       "  df 

Differentiating  the  first  of  (560)  with  respect  to  //,  we  find 

d*dp.oL  d^a       \~dn  d^a  .       d^oT\ 

p  ,  3     =  «2  cos  2«  --y-  4-     -7-  cos  a~  —  mn  sin  «--T-     tan  ^ 
dt  dt        \_dt  dt  dt  J 

Vdn  .  ~\  d,d     •• 

+  I  v/sm  «  +  ^«  cos  a   sec  d-^- 

-f-  2#s  cos  2<*  tan2  <J-^r  +  2«a  sin  2«  tan  d  sec2  ^--77-. 


In  like  manner,  differentiating  the  first  of  (559)  twice  with 
respect  to  yw,  we  find 


a  Jd»a\*    ,  XM^  d^a. 

~  —  —  n  sin-o'  tan  tfl-jr    +  »  cos  or  sec2  dr^r  -^r 

\  at  I  at    at 


»       ^ 
n  cos  <*tan  ^~r  +  n  cos  a?  sec  a-r  -~ 


sin  a  tan  8  sec2  ^ry-)    +  n  sin  «  sec2  ^~4r~' 


2    ~ 


Substituting  in  these  equations  for  ~~,  etc.,  their  values  from 
(562)  and  (563),  then  substituting  in  (566)  these  values,  also 
-— r  and  ~-r  from  (560)  and  (563),  we  have  the  required 


588 


PRA  C  TIC  A  L   ASTR  ONOM  Y. 


§342. 


value  of  the  third  differential  coefficient, 
a  similar  manner.     They  are  as  follows: 


is  found  in 


~dfl 


+  3       ^'  Sin  a 


,n\t.'(m  +  2/it)  cos  a  +  -(/«  +  2/n)«9  COS  za 


+  -  »  -£•  sin  2<x—  2MS  sin*  & 

-f-  I  (2»a  —  nfl  —  6ju.a  -f-  6/u.'2  —  3*ttju.  -(-  3«a  cos  2a)«  sin  a 

+  («"  -^+  »  -^  +  3  ^V)  cos  a  +  6«V  sin  2aJ  tan  8 
-f-  |_6/HM/a  +3  -j*n'  sin  a  -f  (12^  +  3»/)«M;  COS  a 

4"3  w~r  s'n  2a  ~l~  (3W  ~f-  6/a)«2  cos  2a    tan28 
a*  J 

-|-  [(2«a  -|-  6/u./a)w  sin  a  -\-  6«  V  sin  2a  -(-  4«3  sin  a  cos  2a]  tan3  8; 


~*j  SA  "i  s?*i 

£:•  J  =  -  MV  -  (a*  +  3^  sin  a  -  (m*  + 


+  3»//n)»  cos  a  —  w-     sin  a 
at 


(567) 


—  na  sin2  a  cos  a  —  3*  V  sin2  a  —  2  /u.2ju.'  sin2  6 

—  I  6n/jifji'  sin  a  +  -(w  -j-  2/u.)«2  sin  2a  +  3«—  -  sin3  a     tan  8 

2  a/  J 

~  3«2(w  cos  a  -f  n')  sin2  a  tan2  8. 


342.  These  expressions  for  the  third  differential  coefficients 
are  too  complicated  for  use  in  practical  computation.  A 
series  of  tables  is  given  by  Argelander*  by  means  of  which 
that  part  may  be  readily  derived  which  depends  on  preces- 
sion only.  These  tables  are  convenient  when  the  proper 
motion  is  so  small  that  it  may  be  disregarded.  They  are 
given  for  the  epoch  1850,  and  Bessel's  constants  are  employed. 

If  the  third  differential  coefficients  are  required,  they  may 
be  obtained  very  conveniently  by  computing  the  values  of 
the  second  differential  coefficients  for  two  dates  fifty  years 
before  and  after  the  given  one  and  proceeding  according  to 
the  method  of  Art.  50. 

If  we  make/(T)  =         ,  then/(7^-  w)  and  f(T+w)  will 


*See  Untersuchungen  iiber  die  Eigenbewegungen  von  250  Sternen,  p.  145. 


§  343-  EXPANSION  INTO   SERIES.  589 

be  the  values  for  dates  fifty  years  before  and  after  the  date 
T.    Then  the  first  of  (101)  gives 


(568 


the  notation  being  that  of  formula  (101),  and  the  unit  of  time 
being  one  year. 

343.  If  now  we  require  the  precession  formulae  for  any 
given  date,  as  1875.0,  we  obtain  them  by  substituting  for  m 
and  n  the  values  given  by  (549).  m  will  generally  be  ex- 
pressed in  time  and  n  in  arc.  It  will  be  convenient  to  give 
the  formulae  for  the  second  differential  coefficients  the  fol- 
lowing form: 

\dm       m  dn\        dn  I  (da  \  (da          \ 

—  - 1  -f-  —  -I— JU  I  -j-  n  sin  i    I—   -f-  IJL  I  cos  a  tan  o 

(*<\  V 

— -  -hu'fsin  a  sec2  d  -j-  2//X  sin  i"  tan  £; 

</2£~|  _  </«  !/<*#  _     A  .      ,,lda 

1  ^*^ 

,  ^-,  and  /^  will  be  expressed  in  time;  n,  -r ,  and  //  in  arc. 
We  then  have  the  following  formulae  for  1875.0: 


7/~  l~  3s. 07225  +  [0.126115]  sin  a  tan  d  -}-  ju; 

I  - — \-ju\  cos  a  tan 
+[4-81 169] !^-j-Xj  sin  a  sec2  5-|-[4.9866]////'tan  #;   ^-(569) 

~  \=  [1.302206]  cos  a 


5QO  PRACTICAL   ASTRONOMY  §  344. 

The  numerical  quantities  enclosed  in  brackets  are  loga- 
rithms as  usual. 

A  numerical  example  illustrating  the  application  of  the 
foregoing  formulae  is  given  in  Art.  347. 

Star  Catalogues  and  Mean  Places  of  Stars. 

344.  The  various  catalogues  of  stars  which  are  in  use  may 
be  divided  into  two  classes,  viz.,  compilations  and  those  derived 
from  original  observation. 

Among  the  most  important  of  the  first  class  are  the  British 
Association  Catalogue,  Newcomb's  Catalogue  of  1098  Standard 
Clock  and  Zodiacal  Stars,  Boss'  Catalogue  of  '500  Stars,  and  Saf- 
ford's  Catalogue.  These  catalogues  are  of  very  different 
degrees  of  excellence.  The  British  Association  Catalogue 
(often  written  B.  A.  C.)  contains  the  right  ascensions  and 
north-polar  distances  of  8377  stars  reduced  to  the  mean 
equator  of  January  i,  1850.  The  places  of  many  of  these  are, 
however,  not  well  determined,  errors  of  from  5/r  to  10"  in 
north-polar  distance,  and  of  corresponding  magnitude  in 
right  ascension,  not  being  uncommon.  It  is  a  very  conven- 
ient catalogue  for  use  in  preliminary  work,  bnt  the  co-ordi- 
nates of  the  stars  should  be  taken  from  other  authorities 
when  accuracy  is  required. 

The  places  given  in  Newcomb's  and  Boss'  catalogues,  on 
the  other  hand,  have  been  derived  with  great  care  from  all 
of  the  more  reliable  authorities,  and  are  entitled  to  great 
confidence. 

The  following  are  among  the  most  reliable  of  the  other 
class  of  catalogues,  viz.,  those  derived  from  original  observa- 
tion: 

Bradley  s  Observations  reduced  by  Bessel.     Epoch  of  cata- 
logue 1755. 


§  344-  MEAN  PLACES  OF  STARS.  59 1 

Bradley  s  Observations  reduced  by  Auwers.  Epoch  1755. 
Piazzi.  Precipuarum  Stellarum  Inerrantium  Posit iones  Medics. 

Epoch  1800. 

Groombridge.     A  Catalogue  of  Circumpolar'  Stars,  deduced 
from  the  Observations  of  Stephen  Groombridge. 

Epoch  1 8 10. 

•Struve.     Positiones  Medics.  Epoch  1830. 

Argelander.     DXL  Stellarum  Fixarum  Positiones  Medics. 

Epoch  1830. 

Airy.     First  Cambridge  Catalogue.  Epoch  1830. 

Robinson.  Armagh  Catalogue  of  5345  Stars.  Epoch  1840. 
Gilliss.  Observations  made  at  Santiago,  Chili.  Epoch  1850. 
Pulkowa.  Catalogue  in  Vol.  I,  Pulkowa  Observations. 

Epoch  1845. 
Greenwich.     The  various  catalogues  from  observations  at  the 

Greenwich  observatory. 
Radcliffe.     Several  catalogues  from  observations  made  at  the 

Radcliffe  observatory,  Oxford. 
Washington.     Catalogues  derived   from  observations  at  the 

Naval  Observatory,  Washington,  D.  C, 
Besides  these  there  are  valuable  catalogues  published  by 
the  observatories   of    Brussels,  Paris,  Cambridge,  England, 
Cambridge,  U.  S.,  Edinburgh,  Vienna,  and  others. 

These  catalogues  give  the  right  ascension  and  declination 
(or  north-polar  distance)  of  the  stars  referred  to  the  mean 
equator  of  the  date  of  the  catalogue.  Generally  the  data  for 
reducing  the  star  to  the  mean  equator  of  any  other  date  are 
also  given.  These  are  commonly  given  under  the  headings 
precession  and  secular  variation  ;  the  proper  motion  is  some- 
times given  when  its  value  is  known. 

The  quantities  called  precession  are  simply  the  values  of 

-  and  -jr  for  the   date  of  the  catalogue,   precession  only 


592  PRACTICAL  ASTRONOMY.  §  345. 

being  considered.      The  secular  variations  are  the  changes 
which  take  place  in  these  quantities  in  100  years  ;  i.e.,  the 


values  of  100  -      and  100  --. 

-y 


Let  /«  =  the  annual  precession  in  right  ascension  =  -y-; 


72 

sa  =  the  secular  variation  —  100  -JTT  ; 

at 

«0  =  the  right  ascension  for  epoch  T9  the  date  of  the 

catalogue  ; 
a  =  the  right  ascension  for  epoch  T  -\-  1. 


Then  «  =  «.  +  //.+  ;  -'-) (570) 


The  decimation  will  be  given  by  a  similar  process.  If  proper 
motion  is  given,  this  must  also  be  included  in  formula  (570). 
In  some  catalogues  the  proper  motion  is  included  with  the 
precession,  when  this  is  generally  given  under  the  heading 

da       jdS 
annual  motion,  and  it  corresponds  exactly  to  -j-  and  -j-  given 

by  formulae  (565). 

345.  When  a  star's  place  is  required  with  extreme  accu- 
racy it  should  be  sought  for  in  as  many  original  authorities 
as  may  be  available,  and  the  values  of  the  co-ordinates  given 
by  the  various  catalogues  combined  by  the  method  of  least 
squares  to  determine  the  most  probable  values  of  these  co- 
ordinates with  the  proper  motion.  There  are  different 
methods  for  working  out  the  details  of  this  process,  the  fol- 
lowing being  perhaps  more  frequently  employed  than  any* 
other : 

Suppose  we  require  the  mean  place  for  1875.0,  together 
with  proper  motion.  If  the  star  has  been  well  observed  at 


§  345-  MEAN  PLACES  OF  STARS.  593 

epochs  separated  by  a  considerable  interval,  the  latter  may 
be  determined  ;  otherwise  not. 

We  first  derive  the  approximate  right  ascension  and  decli- 
nation for  1875.0  by  reducing  to  that  date  the  place  as  given 
in  one  or  more  of  the  best  modern  catalogues,  using  for  this 
purpose  the  annual  motion  and  secular  variation  of  the  cata- 
logue. For  this  preliminary  place  the  Greenwich  catalogues 
will  generally  give  a  value  of  the  right  ascension  within  s.2 
or  8.3,  and  of  the  declination  within  2"  or  3"  of  the  truth. 

c  da    dd  d*a         ,  d*$ 
We  then  compute  accurate  values  of  -T-,  -3-,  -^-,  and  -j^ 

for  1875.0  by  formulae  (569) ;  and  if  great  precision  is  required, 

d'a       .  d*$ 

—TP  and  -7-5-,  as  explained  in  Art,  342.  Our  assumed  co-ordi- 
nates are  then  to  be  corrected  by  comparing  them  with  the 
places  given  in  the  various  catalogues.  For  this  purpose 
the  assumed  right  ascension  and  declination  are  reduced  to 
the  date  of  each  catalogue. 

Let  <xl  =  the  assumed  right  ascension  for  1875.0; 

ar/  =  the  value  of  al  reduced  to  the  epoch  of  catalogue, 

1875  -  /; 

o?2  =  right  ascension  given  by  catalogue ; 
/*  =  the  annual  proper  motion. 

The  difference  (<ara  —  or/),  supposing  for  the  present  #3  to 
be  free  from  error,  will  consist  of  two  parts,  viz.,  the  error 
\  in  the  assumed  value  of  al  and  the  change  due  to  proper 
motion  in  the  interval  t.     Therefore 

x  -  »t  =  K  -  o (571) 

is  an  equation  for  determining  the  proper  motion  //  and  the 
correction  to  the  assumed  right  ascension  x.  Each  catalogue 
will  give  us  an  equation  of  this  form ;  trom  these  the  most 


594 


PRACTICAL  ASTRONOMY. 


§346. 


probable  values  of  x  and  /*  are  derived  by  least  squares.  A 
similar  series  of  equations  will  also  be  obtained  for  the  Decli- 
nation. 

346.  The  above  is  an  outline  of  the  method ;  practically  it 
is  much  complicated  by  the  fact  that  the  different  catalogues 
are  of  very  different  degrees  of  accuracy,  and  in  the  same 
catalogue  the  weight  will  depend  on  the  number  of  observa- 
tions made  on  the  star.  It  is  impossible  to  give  any  infallible 
rule  for  the  assignment  of  weights ;  practically  much  must 
depend  on  the  judgment  of  the  investigator.  In  general, 
however,  the  more  recent  catalogues  are  entitled  to  much 
greater  weight  than  the  older  ones.  Methods  and  instru- 
ments are  constantly  improving,  and  in  consequence  a  much 
higher  precision  is  possible  now  than  was  the  case  a  hundred 
years»ago.  The  old  catalogues  are,  however,  indispensable 
in  investigation  of  proper  motion. 

The  following  table  shows  the  weights  assigned  by  New- 
comb  to  the  different  authorities  employed  in  deriving  the 
right  ascensions  of  the  catalogue  referred  to  above : 


Number  of  Observations. 

i 

2 

3 

4 

5 

7 

10 

15 

20 

25 

30 

40 

5° 

60 

So 

IOO 

i 
4 

i 
8 

6 

Bessel's  Bradley 

f 

i 

| 

I 

! 

I 

| 

I 

| 
i 

2 

I 

* 

1 

2 

I 

i 

i 
I 

I 

2 

i 

i 

3 

2 

* 

I 
I 

*, 

2 

* 

2 

I 

4t 

2 

* 

2 
I 

5 

3 

1 
3 
I 

5 

3 

i 
3 
i 
6 

4 

i 

3 

i 
6 

4 

f 
4 

I 

S 

Auwers'  Bradley  
Piazzi                                       

Struve,  1825     '.  
Argelander,  1830  

Pond  

Airy,  Cambridge,  1830  

" 

Airy,  Greenwich,  1840  

I 

i 
\ 

i 

2 

I 

I 

•2 

I 

i 
3 

t 

k 

4 

i 

k 

5 

i 

I 

7 

2 

I 

7 

2 

2 

8 

2 

2 

IO 
2 

3 

15 
3 

3 
20 

3 

4 

20 

4 

4 

25 
4 

5 
25 
5 

3 

Radcliffe,  1845  

Airy,  Greenwich,  1845  

8 

10 

\t 

if 

20 

2.5 

25 

3 

Airy,  Greenwich,  1850  
Pulkowa,  1850.  ...       
Airy,  Greenwich,  1860  . 

i 

2 

2 

? 

4 

ft 

7, 

I 

Yarnall,  Washington,  1860  
Airy,  Greenwich,  1864  
Engleman,  Leipzig,  1866  

11 

« 

" 

" 

i. 

| 

A  iry,  Greenwich,  1870  
Washington,  1870  

" 

4 

§347- 


MEAN  PLACE    OF  STAR   B.A.C.    2786. 


595 


Boss  gives  a  similar  table  of  weights  for  the  declination 
equations.  See  Report  of  the  U.  S.  Northern  Boundary 
Commission,  p.  566. 

If  an  approximate  value  of  the  proper  motion  is  also  known 
it  may  be  employed  in  computing  the  differential  coefficients 
by  formulas  (569),  when  we  shall  have  in  equation  (571),  in- 
stead of  /-/,  the  correction  to  the  assumed  value  of  ^,  viz.,  A  p. 

Example. 

347.  For  the  purpose  of  illustrating  the  foregoing  formulae  and  methods  let 
us  derive  the  mean  co-ordinates  and  proper  motion  of  the  star  B.  A.  C.  2786* 
for  the  epoch  1875.0.  The  following  tabular  statement  shows  the  values  of  the 
co-ordinates  given  by  the  various  authorities  consulted.  It  probably  explains 
itself  sufficiently. 


i 

og 

rt 

*O  C 

rt 

Catalogue. 

J2  3 

W  w 

ij 

0. 

"o  5 

Catalogue 
Right 
Ascension. 

li 

a> 

ifi 

§1 

Catalogue 
Declination. 

a— 

§o.s 

.2 
o  **^ 

Jo! 

.  O 

W~ 

s 

* 

Bradley  

1755 

5 

8h    sm    8s.  03 

. 

4 

27°  59'  22".6 

Piazzi 

1800 

• 

8      7     53  .15 

8 

27      51     13     .O 

Gould's  D'Agelet.. 

1800 

1783.3 

8      7     53  -3 

1783-3 

27      5I     22     .0 

Weiss'  Bessel  

1825 

1826.2 

2 

8      9     24.70 

1826.2 

2 

27    46    34    .0 

Argelander  

1830 

8 

8      9     43  -43 

8 

27    45    4°     3 

Taylor  

l835 

6 

80       i  .96 

4 

27    44    46    .86 

Armagh  

1840 

1830.2 

i 

8      o     19  .99 

1853.3 

5 

27    43    43    -31 

Brussels  

1856 

1856.1 

6 

8      i     18  .56 

1856.2 

i 

27    40   49    -37 

'•        

1858 

1858.1 

4 

8      i     25  .98 

1858.1 

4 

27    40   26    .8 

14 

1860 

1860.  1 

i 

8      i     33  -M 

1860.1 

2 

27    4«     5    -5 

Cape  of  Good  Hope. 
Greenwich  

1860 
1860 

1857.1 

1857.7 

2 

8 

8      i     33-38 
8       i     33  .28 

1857.1 
1857.7 

8 

27    40     4    .37 
27    40     4    .12 

Radcliffe     

1860 

1855.0 

5 

8    ii     33  .29 

1856.3 

7 

27    40      4    .2 

Greenwich 

1864 

1863.7 

6 

8    ii     47  .88 

1863.7 

10 

27    39    18    .96 

1868 

1868.2 

3 

8    12       2  .53 

1868.2 

9 

27    38    33    -76 

"            

1869 

1869.2 

8      12          6  .22 

1869.2 

i 

27      38     23      .TO 

** 

1870 

1870.2 

6 

27      38      II      .63 

u            

1871 

1871.2 

6 

27    38     o   .30 

u 

1872 

1872.2 

V 

27    37   40    .02 

Washington  

1872 

1872.2 

3 

8    12     17  .08 

1872.2 

3 

27    37    48    -5 

We  first  require  an  approximate  value  of  the  star's  place  for  1875.0,  which  we 
may  readily  derive  from  the  four  catalogues  which  give  the  co-ordinates  for 
1860.0,  viz.,  Brussels,  Cape  of  Good  Hope,  Greenwich,  and  Radcliffe.  Thus  we 
find 

1860     a  =  8h  nm  338.27; 


*  This  is  the  number  of  the  star  in  the  British  Association  catalogue. 


PRACTICAL   ASTRONOMY.  §  347. 

For  reducing  these  to  1875  we  take  from  the  Greenwich  catalogue  the  follow- 
ing quantities  : 

In  right  ascension,       precession  =  -f-  3".66i; 
secular  variation  =  —     .017. 

In  declination,  precession  =  —  io'f.8g; 

secular  variation  =>  —         .44; 
proper  motion  n'  =  —        .38, 
Therefore 

1875      a  =  8h  nm  338.27  -f  15(3.661  —  7.5  X  .00017)  =  8h  i2m  28M7; 

d  =  27°  40'  4". 5    -f  is(—  lof'.Bg  —  7-5  X  .0044  -.38)  =  27°  37'  15". o 

We  may  reasonably  expect  these  to  prove  very  close  approximations  to  the 
final  values.     With  these  values  of  a  and  d,  and  the  above  value  of  ju',  we  next 

da  dS     d^a          d^d 
compute  — -,  — -,  —jj9,  and^2"  by  (569).     This  computation  is  given  in  full. 

Constant  =       0.126115     Constant  =     4.63380*        Constant  =     4.63380* 
sin  a  =       9.923012       da  d§ 


log  =       9.767837  log  =     5.19706*  log  =     5.67348 

Nat.  No.  =       0.58592       Nat.  No.       —  .000015  7  Nat.  No.  =+.000047  15 
m  =        3.07225        Constant  =     5.98778          Constant  =     7.16387* 

da  da  da 

JLI  =        3.65817 }-  M  =       .56326  —  -  -}-  ju  =       .56326 

»    log  n  =       1.302206  cos  a  —     9.73748*  sin  a  =     9.92301 

cos  a  =       9.737476*          tan  d  =     9.71871 

log  =        1.039682*  log  =     6.00723*  log  =     7.65014* 

Nat.  No.  =  —  10.95676     Nat.  No.  =—  .000  101  7  Nat.  No.  =  —  .004468  30 
//'==  —       .38  Constant  =     4.81169 

—  =   —  11.3368  — -  +  JU'  =      I  06881*  -j£  =  —  .004  421  2 

sin  a  =     9.92301 
see2  d  =       .10510 

log  =     5.90861* 
Nat.  No.  =—  .000  08 1  o 
Constant  =-f-  .000  032  2 


-yy    =—     .000  166  2 


§  347- 


MEAN  PLACE    OF  STAR   B.A.C.    2786. 


597 


For  determining  the  third  differential  coefficients,  we  find  for  the  dates  1825 
and  1925  respectively: 


%  1825     -^5   =  -  .000  164  5;        —  =  -  .004  471  5. 
1925    -~  =  -  .000  167  9;       —  =  -  .004  370  o. 
We  therefore  find,  by  (568), 

— —  =  —  .000  ooo  034;        -7-^  =  -f-  .000  ooi  014. 

Substituting  the  above  values  of  the  differential  coefficients  in  Maclaurin's 
formula,  and  making  /  minus,  since  we  shall  want  to  apply  it  to  dates  previous 
to  1875,  we  have 

a  =  a0  —  /[3s. 658 17-]- /(.ooo  083  i  —  /.ooo  ooo  006)]; 
8  =  d0  +  /[n".3368  —  /(.oo2  211     -f-  /.ooo ooo  17)] 

By  means  of  these  formulae  we  next  reduce  the  above  assumed  right  ascen- 
sion and  declination  to  the  epoch  of  each  of  the  authorities  where  our  star  is 
found.  • 

The  differences  between  these  computed  values  and  the  observed  values  are 
given  in  the  following  table.  The  "  correction  for  //'"  there  given  is  applied 
to  those  catalogues  where  the  epoch  of  observation  differs  considerably  from 
the  epoch  of  the  catalogue.  For  example,  Gould's  D'Agelet:  The  mean  epoch 
of  observation  is  1783;  the  catalogue  places  are  given  for  1800.  We  have  as- 
sumed //'  =  —  ".38,  which  in  17  years  produces  a  change  in  3  of  —  6". 46.  This 
is.  in  this  case,  the  "correction  for  /*'•" 


t 

RIGHT  ASCENSION. 

DECLINATION. 

£ 

£ 

AUTHORITY. 

«   M 

Jc 

0  -  C. 

C   .! 

O   03 

cfl  c 

J 

io 

O  -  C. 

3 

Jj  rt  J3  b 

be 

Sj 

be 

u**"  -  " 

£ 

**jjl 

£ 

n. 

*. 

se£ 

r- 

S 

cSl"" 

n. 

V. 

I 
2 

Bradley 

1800 

5 

7 

.5 

+.03 
—  .19 

+.04 
-.18 

T755 
1800 

4 

8 

2 

I 

+  -25 

-    -53 

Piazzi       

3 

Gould's  D'Agelet  .  .  . 

1783    ' 

—  .04 

1783 

os 

-6".46 

+2-79 

+2.99 

4 

Weiss'  Bessel  

1826      2 

.1 

—  •35 

—  .34 

1826 

2 

i 

+     .38 

—  1.91 

-1.67 

5 
6 
7 

Argelander  
Taylor   
Armagh  

1830    8 
,835    6 
1830    i 

2.O 

•5 
•'j 

+s 

—.04 

+.26 
—  -°3 

.830 
1835 

8 
4 

s 

2    0 
2 

+4   -94 

+1.94 
-    .82 

—   .11 

+2.19 
—    -54 

8 

Brussels 

1856    6  j 

1856 

T  ) 

*• 

9 

18581  4f 

2.O 

-.07 

-.06 

1858 

4 

.8 

+   -14 

+  -42 

o 

44 

i860        T    \ 

1860 

0    } 

i 

2 

3 

Cape  of  Good  Hope. 
Greenwich   
Radcliffe       ,       .     . 

i860 

1860 
1860 

8 

—  o 

+  01 

fi 

1860 
1860 
1860 

2 

8 

•7 

2  .O 

-    .18 
-    -43 

+    .10 

-    -IS 

4 

Greenwich  

18^ 

| 

1864 

—       8 

1868 
1869 

l\ 

3-0 

—  .01 

+.01 

1868 
1869 

1 



7 

1870 

6!- 

2.0 

-    .06 

+  -23 

8 

6  I 

19 

,  1 

20 

Washington 

1872 

3 

1.0 

—  .11 

-.09 

1872 

3 

8 

—    -49 

-     19 

59S  PRACTICAL  ASTRONOMY.  §  348. 

The  weights  have  been  assigned  in  accordance  with  the  systems  of  New- 
comb  and  Boss  for  the  most  part. 

The  quantities  n  are  now  the  absolute  terms  of  the  system  of  equations  of 
condition  of  the  form 


a  —  tn  =n)          and          V/(J<$  —  tAjn1  =  n). 

From   these  we  derive   the  following  normal  equations  in  the  usual  manner, 
with  the  values  of  the  unknown  quantities: 

^  2i.25oz/cc  —  4.045//  —  —  .304; 

—    4.045^0:  +  i. 365^  =+  .055; 

Aa  —  -  .015      ±  .0197; 

/*  =  —  .00005  ±  .00078. 

11.750^/5  —  2.4i6^//'  =  —  3.263; 

—     2. 4I6//5  -|-     .987.41*'  =  +     .615; 

Ad  =  —    .301      ±  .122; 
AIJL'  —  —    .00114  ±  .00420. 

Applying  these  corrections  to  the  assumed  values  of  a,  d,  and  ju',  we  have 
finally,  as  the  most  probable  values, 

a  =  8h  12™  28M55  ±  .0197;         //  =  —  '.00005  ±  .00078; 
d  =  27°  37'  14". 70  ±  .122;          //'  =  —  ".3811     ±  .0042. 


Nutation. 

348.  Nutation  has  already  been  defined  as  the  name  applied 
to  the  periodic  part  of  the  precession.  The  components  of 
the  attractive  force  of  the  sun  and  moon,  which  tend  to  draw 
the  equator  into  coincidence  with  the  ecliptic,  are  not  con- 
stant with  respect  to  either  of  those  bodies.  The  component 
has  a  maximum  value  when  the  attracting-  body  is  in  the 
plane  passing  through  the  earth's  axis  and  perpendicular  to 
the  ecliptic,  and  it  is  zero  when  the  body  is  in  the  plane  of 


§  349-  NUTA  TION.  599 

the  equator.  The  orbit  of  the  moon  and  apparent  orbit  of 
the  sun  are  ellipses,  so  that  the  distances  of  these  bodies  from 
the  earth  are  constantly  changing.  The  angle  between  the 
plane  of  the  moon's  orbit  and  the  equator  is  variable;  so  in  a 
less  degree  is  that  between  the  equator  and  ecliptic,  or  ap- 
parent orbit  of  the  sun.  All  of  these  circumstances  produce 
periodic  terms  in  the  movement  called  precession. 

It  will  be  seen  that  the  law.  or  laws  governing  this  matter 
are  intricate  and  difficult  to  investigate  ;  their  discussion  be- 
longs to  the  department  of  Physical  Astronomy.  Various 
investigators  have  given  more  or  less  attention  to  the  deter- 
mination of  the  constants  which  enter  into  the  formulas;  the 
values  which  are  most  extensively  employed  at  present  are 
those  of  Peters. 

349.  Since  nutation  is  simply  a  motion  of  the  equator,  the 
ecliptic  remaining  unchanged,  it  follows  that  it  will  produce 
no  effect  upon  the  latitudes  of  stars.  The  longitudes  will  be 
changed,  also  the  obliquity  of  the  ecliptic. 


Let  A\  and  4(&  =  the  nutation  in  longitude  and  obliquity 
respectively. 

Then,  according  to  Peters,  for  1800.0: 

^A  =  —  17".  2405  sin  Q+".  2073  sin  2  Q—  ".  2041  sin  2  C-f-".o677sin(([  —  r")"| 
-   i".26g2  sin  2  Q  -(-".1279  sin  (©  —  -T)  —".0213  sin  (0  -J-.T)  ; 

os  2©       57  ' 


4a)=       9".  2231  cos  Q  —  ".0897  cos  2  Q  -|-"-o886cos  2  C  +  ''.5509  cos  2© 
+     ".0093  cos  (0+r).  J 

Where  Q  =  the  mean  longitude  of  the  ascending  node  of  the  moon's  orbit  ;* 
£  =  the  moon's  true  longitude  ; 
0  =  the  sun's  true  longitude  ; 

F  =  true  longitude  of  the  sun's  perigee  ;  % 

r"  =  true  longitude  of  the  moon's  perigee. 

*That  is,  the  point  where  the  moon  passes  from  below  the  ecliptic  to  above. 


6OO  PRACTICAL  ASTRONOMY.  §  350. 

The  coefficients  of  the  above  formulae  vary  slowly  with 
the  time,  so  that,  according  to  Peters,  the  values  for  1900 
will  be 


.  ".2073  sin2Q—".204isin2C+".  0677  sin(C—r')>)  * 

—  i".2693  sin  2©  +".1275  sin  (O  —  -T)  —  ".0213  sin(0  -\-  F)  ; 
4oo'=-{-  9".  2240  cos  £—  ".0896003  2  Q  -p;.  0885  cos  2(1  +  ".5506  cos  2Q  I 
-f     ".0092  cos  (0+T).  J 

The  numerical  values  of  ^/A,  and  the  true  obliquity, 
=  GO  -f-  ^c0,  are  given  in  the  ephemeris  for  every  tenth  day 
throughout  the  year.  A\.  is  there  called  the  equation  of  'the 
equinoxes,  'and  is  additive  algebraically  to  the  longitude  re- 
ferred to  the  mean  equinox  in  order  to  obtain  the  longitude 
referred  to  the  true  equinox. 

350.  To  determine  the  nutation  in  right  ascension  and  declina- 
tion. Since  the  terms  of  the  formulae  are  always  small,  a 
sufficiently  accurate  result  will  be  obtained  by  neglecting  the 
squares  and  higher  powers  of  these  quantities.  In  other 
words,  we  may  employ  differential  formulae,  viz., 


da  da 

Aa  —  -rr^A  4-  3— 
d\.         r  doo 

•    •    •    •    (574) 


For  the  values  of  the  differential  coefficients  we  employ 
the  equations  obtained  by  applying  the  general  formulae  of 
trigonometry  to  the  triangle  formed  by  joining  the  poles  of 
the  equator  and  ecliptic  with  each  other  and  with  the  star. 

*  In  No.  2387,  Astronomische  Nachrichten,  Oppolzer  gives  formulae  for  these 
quantities  carried  out  so  as  to  include  all  terms  which  are  appreciable  in  the 
fourth  decimal  place. 


§  350.  NUTATION.  6CI 

In  Fig.  72,  P  is  the  pole  of  the  ecliptic,  P'  of  the  equator, 
5  any  star. 

PP'  =  GO,        PS  =  90°  -  ft,         PfS  =  90°  -  d, 
SPP'  =  90°  --  K,        SP'P=  90°  +  a. 

Therefore 

COS  d  COS  a  =  COS  /?  COS  A  ;  ^ 

cos  d  sin  a  =  cos  /?  sin  A  cos  G?  —  sin  ft  sin  &? ;  I  (575) 

sin  tf  =  cos  ft  sin  A  sin  GO  -f-  sin  /J  cos  GO.  J  FlG  ?2 

Differentiating  these  equations,  considering  ft  as  constant, 
since  it  is  not  affected  by  nutation, 

cos  d  sin  ada  -j-  cos  a  sin  ddd  =  cos  /?  sin  A*/A  ; 

cos  5  cos  ada  —  sin  or  sin  ddd  =  cos  /?  cos  A  cos  aodA.  I (576) 

—  (cos  ft  sin  A  sin  ca-j-sin  ft  cos  (ajdoo;  \ 

cos  &/£  =  cos  fi  cos  A  sin  (»</A  +  (cos  ft  sin  A  cos  &?— sin  ft  sin  eo)</o9.  J 

From  the  second  and  third  of  (575)  we  derive 

cos  ft  sin  A  =  cos  tf  sin  «  cos  GO  -f-  sin  d  sin  GO. 
Reducing  (576)  by  this  and  the  first  of  (575),  we  have 

cos  d  sin  ada  -}-  cos  a  sin  SdS  =  (cos  5  sin  a  cos  <»  -f~  sin  d  sin 

s  GJ 
+ 

From  these  we  derive 


cos  d  sin  ada  -{-  cos  a  sin  ddS  =  (cos  5  sin  or  cos  GO  -\-  sin  d  sin  a?yA;       j 
cos  d  cos  m/a:  —  sin  a  sin  6V£  =  cos  S  cos  a  cos  aodX  —  sin  6V(»  ;  r  (577) 

dd  =  cos  a  sin  aodh.  4-  sin  or</(».  ) 


da  dd 

—  =  cos  GO  -f-  sin  cy  sin  «  tan  tf  ;  ^-  =  cos  a  sm  GO; 

dn  ds         ;  (578> 

-7-  =  —  cos  «  tan  o  ;  3—  —  sin  «. 

</Gi9  </&? 


6O2  PRACTICAL  ASTRONOMY.  §  35O. 

Substituting  (572)  and  (578)  in  (574),  we  have* 

Acc=— (15". 8148  -}-  6". 8650  sin  a  tan  5)  sin  Q  —  9.2231  cos  or  tan  <5  cos  Q 

15   .8321       6  .8683  9.2240 

-}-      (.1902  -j-  .0825  sin  a.  tan  5)  sin  2  Q  -f-  .0897  cos  a  tan  d  cos  2  Q 

.0895 

—  (.1872  -}-  .0813  sin  a  tan  5)  sin  2(£  —  .0886  cos  a:  tan  d  cos  2(£ 

.0812  .0885 

-f-      (.0621  -f-  .0270  sin  a  tan  <5)  sin  (C   —  -O 
-j-  .000  154  cos  2a  tan*#  sin  2  Q  —  .000  160  sin  2a  tan2£  cos  2  Q 

—  (1.1642  -j-  .5054  sin  a  tan  <5)  sin  2©  —  .5509  cos  a  tan  6"  cos  2Q 

1.1644         5°52  55°6 

-{-      (-II73  +  .0509  sin  a  tan  5)  sin  (0  —  JT) 

1170        0507 

— (.oi95-|-.oo85  sin  a  tan  5)  sin(0  -}-O— .0093  cos  a  tan£  cos(0-f.T) ; 

0092 

•(579) 
Jd=—  6". 8650  cos  a  sin  Q  -|-  9". 2231  sin  a  cos  Q 

6  .8683  9  .2240 

-}-  ''.0825  cos  a  sin  2Q  —  ''.0897  sin  a  cos  2Q 
.0895 

—  .0813  cos  a  sin  2(£  -f-   .0886  sin  a  cos  2(£ 
0812  0885 

-}-   .0270  cos  a  sin  ((£  —  F') 

—  .000  077  sin2«  tan5  sin2  Q  —  (.oooo23-}-.ooo  080  cos2a)  tan5  cos2  Q 

—  .5054  cos  a  sin  20  -f-      .5509  sin  a  cos  2O 
.5052  5506 

-j-      .0509  cos  a  sin  (O  —  -O 
0507 

—  .0085  cos  a  sin  (0  -j-  P)  +  -0093  sin  or  cos  (Q  -J-  F). 

.0092 

In  case  of  those  coefficients  which  change  appreciably 
during  the  century  the  value  for  1900.0  is  written  below  that 
for  1800.0. 

Tables  have  been  prepared  for  facilitating  the  computation 
of  the  above  formulae,  but  they  do  not  require  special  con- 
sideration here.  For  our  purposes  the  necessary  corrections 

*  See  Peters'  Numerus  Constans  Nutationis.     Also  Astronomische  Nachrichten, 
No.  486. 


§  3  5 1  ABERRA  TION.  603 

are  computed  in  a  simple  manner,  as  explained  in  Articles 
354  and  following. 


Aberration. 

351.  Aberration  is  an  apparent  displacement  of  a  star's 
position,  resulting  from  the  circumstance  that  the  velocity  of 
light  is  not  infinitely  great  in  comparison  with  the  velocity 
of  the  earth.  Two  essentially  different  classes  of  phenomena 
result  from  this  cause  : 

First.  The  observer,  who  must  partake  of  all  the  motions 
of  the  earth  itself,  does  not  see  the  object  in  its  true  posi- 
tion, since  the  observed  direction  of  a  ray  of  light  is  deter- 
mined not  by  the  absolute  direction  of  motion  of  the  undu- 
lations coming  from  the  object  to  the  eye,  but  by  the  relative 
motion  with  respect  to  the  observer.  This  apparent  change 
of  position  is  called  the  aberration  of  the  fixed  stars. 

Second.  The  observer  does  not  see  the  body  in  its  true 
position  at'the  instant  when  the  light  enters  the  eye,  but  in 
the  position  which  it  occupied  when  the  light  left  the  body. 
This  is  called  planetary  aberration.  This  latter  we  shall  not 
have  occasion  to  consider,  as  it  belongs  to  another  depart- 
ment of  astronomy. 

The  aberration  of  the  fixed  stars  is  determined  by  the  velo- 
city and  direction  of  the  motion  of  the  point  on  the  earth's 
surface  occupied  by  the  observer.  Of  these  motions  there 
are  three,  viz.,  that  due  to  the  diurnal  revolution  of  the  earth 
on  its  axis,  to  its  annual  revolution  about  the  sun,  and  to  the 
motion  of  the  earth  with  the  sun  in  space. 

The  first  of  these  motions  produces  diurnal  aberration, 
which  has  already  been  considered  so  far  as  is  necessary  for 
our  purposes.*  The  last  motion  it  is  not  important  to  con- 

*  See  Articles  173  and  303. 


604  PRACTICAL   ASTRONOMY.  §  352. 

sider,  as  it  affects  the  place  of  the  star  by  a  constant  quantity  ; 
further,  it  is  not  sufficiently  well  determined  for  the  purpose, 
even  if  it  were  desirable  to  consider  it.  It  only  remains, 
therefore,  to  investigate  the  change  produced  by  the  earth's 
motion  in  its  orbit,  called  annual  aberration. 

352.  Let  the  velocity 'of  the  ray  of  light  coming  from  a 
star  and  of  the  earth  respectively  be  considered  with  respect 
to  three  co-ordinate  axes,  the  equator  being  the  plane  of  X,  Y. 

dx   dy  dz 
Let  -7-,  Hj,  -7-  =  the  components  of  the  earth  s  velocity  in 

the    direction  of    the    three   axes  (the 
measure  of  the  velocity  being  the  space 
passed  over  in  i  second) ; 
k  =  the  velocity  of  light  =  distance  traversed 

in  I  sidereal  second  ; 
a  and  d  =  true  right  ascension  and  declination  of 

the  star ; 
=  the  co-ordinates  of  point  where  the  ray 

of  light  pierces  the  celestial  sphere ; 
£,  77,  £  ='  components  of  velocity  of  the  ray  of  light 
in  direction  of  the   three   co-ordinate 

axes. 
Then 

£  =  —  k  cos  d  cos  «;     77  =  —  k  cos  d  sin  a;     Z  =  —  £sin  #.(580) 

These  are  minus,  since  the  light  moves  in  a  direction  oppo- 
site to  that  in  which  the  star  is  seen. 

Let  the  same  symbols  affected  by  accents  represent  the 
corresponding  quantities  affected  by  aberration.  Then 

£'=—£' cos  £' cos  a';   rf=—  #cos  tf'sin  a;   <?'=— £'sin  £'.(581) 

a'  and  $'  are  then  the  apparent  right  ascension  and  decli- 
nation of  the  star,  and  £' ',  rft  2'  are  the  components  of  the 
velocity  relatively  to  the  earth. 


§  3  5  2  c  A  BERRA  TION.  605 

Since  then  the  relative  velocities  are  equal  to  the  differ- 
ences of  the  actual  velocities, 


k'  COS  ^'  COS  a'  —  k  COS  ^  COS  a  -f-  — ; 

dy 
k'  cos  d'  sin  af  =  k  cos  £  sin  «  -f-  ~; 

dft 

k'  sin  #'  =  k  sin  tf  +  -£. 


-  -    (582) 


/ 
Let  —  =  x.   Then  we  readily  derive  from  these  equations 

K 

the  following  : 


xcos 


i  [""<**  </y        "1 

d'sin(a'  —  <3:)=—  ^    —  -  sin  a  —  -g-  cos  a    ; 

if^  ,  ^v         "1 

osCa'  —  a)=cos<5  +  -|  —-cos  a-}-—  -  sin  a     ; 


'  —  6)  =—  7    —-sin  d  cos  «:-)-  -=--  sin  d  sin  a  --  --cos  5 


I  \  dx    .     K  .    dv     .     .    .  afe 

—  sin  o  cos  or+  — -  sin  o  sin  a -- 

>6     <//  '    <//  ^ 

-         ^(583) 
i      ^Ls}n  a_(fy_cosa  I  tan^- 


-  5)  =i  +  |    ^  cos  *  cos  or  +  -^-  cos  5  sin  a  +  ~  sin  5  I 


dv 


The  first  two  of  these  are  exact ;  the  last  two  are  exact  to 
terms  of  the  second  order  inclusive. 

Dividing  the  first  by  the  second  and  the  third  by  the 
fourth,  we  have,  neglecting  terms  of  the  third  and  higher 
orders, 


606  PRACTICAL   ASTRONOMY.  §  352. 

ct  —  a  =  -  \  sec  8\  ~  sin  a  -  -~  cos  a 
k  \_dt  dt 


dx  .  dy  . 

— -costt-J-^-smtt    ; 


(584) 


\Vdx    .  .    dy   .  dz 

d  —  d  =  —  -,    — ,-  sm  d  cos  a  -f-  —,-  sm  o  sin  a  —  —  cos  o  I 
k  [_  dt  at  dt 

I    \~dx     .  dy  T 

— -  sm  a ~-  cos  a      tan  S 

2/t*\    dt  dt 

L—  —I 

,     i  \~dx  ,    dy    .  dz 

+  k\Jt  sm  *  cos  a  +  *  sm  *  sm  a  ~  7/ cos  d\ 

X      ~  cos  ^  cos  a  -\-  -jj  cos  d  sin  a  -}-  —  sin  d    . 

Let  7?  =  the  radius  vector  of  the  earth; 

O  =  the  sun's  geocentric  longitude ;  then  —  O  =  the 

earth's  heliocentric  longitude; 
GO  =  the  obliquity  of  the  ecliptic. 

Then  x,  y,  z  being  the  earth's  rectangular  co-ordinates, 

x  —  —  R  cos  O;  y  =  —  JRsin  O  cos  GO;  z—  —  y?sinOsin  GO.  (585) 

From  these  we  have 


dx        0^0    .  dR 

—  =  R  —=-  sin  O  —  -rr  cos  O; 
dt  dt  dt 

dy         &^d®  dR   . 

~-  =  -%R  —j-  cos  O  cos  GO  —  -j-  sm  O  cos  c»; 

dt  dt  dt 


dz  ^dQ  .  dR    .  . 

-  —  —  R  -—-  cos  O  sin  GO  —  -j-  sin  O  sm  GO. 
dt  dt  dt 

By  means  of  these  equations  we  have  the  values  of  af  —  a 
and  d'  —  d  in  terms  of  the  sun's  distance  and  longitude,  but 
they  are  not  in  a  convenient  form  for  practical  application 
unless  we  are  satisfied  with  an  approximation  obtained  by 


§353-  ABERRATION.  6o/ 

regarding  the  earth's  orbit  as  a  circle  and  the  motion  uniform. 
In  this  case  we  make  -j-  =  o  and  -g-  =  the  mean  apparent 

angular  velocity  of  the  sun  in  longitude. 

353.  The  true  velocity  of  the  earth  in  any  part  of  its  orbit 
may  be  taken  into  account  as  follows  :  The  orbit  being  an 
ellipse,  its  polar  equation  will  be 


r, 

= 


a  being  the  semi-major  axis,  e  the  eccentricity,  and  —  (0  —  JT) 
the  angle  between  the  major  axis  and  radius  vector  measured 
from  the  perihelion  (O  and  P  having  the  same  significance 
as  in  Art.  349). 


Let  F  =  the  area  of  the  ellipse  =  nc?  Vi  —  e*\ 

T  =  the  time  of  one  revolution  of  the  earth  =  one 

sidereal  year; 
df  —  an  element  of  area  between  two  consecutive  radii 

vectores; 
dt  =  time  required  to  describe  df. 

Then  by  Kepler's  first  law,  viz.  —  the  areas  described  by  the 
radius  vector  are  proportional  to  the  times  —  we  have 


_ 

T   -  dt1  ~T~      ~  2 

since  the  element  of  area  df  =  i#W(O  —  JT)  = 
Therefore 


608  PRACTICAL  ASTRONOMY. 

By  differentiating  (587)  we  find 

dR       (2n\       a 

=  (-—}— =  e  sm  (O  — 

dt        \  T)  Vi  —  e* 


§353- 


(590) 


But   -j^is  equal  to  the  mean  angular  velocity  of  the  earth 

in  its  orbit  about  the  sun  ;  or,  what  is  the  same,  the  apparent 
angular  velocity  of  the  mean  sun  about  the  earth.  Calling 
this  velocity  n,  we  have,  from  (586),  (589),  and  (590), 


dx_ 

~Hi  ~~  ''    y 

dy_=_ 
dz 


an 


Vi  - 
an 


(sin  O  +  e  sin  7"1); 


cos  GO  (cos  O  +  e  cos  F); 


sin  co  (cos  O  +  e  cos  F). 


•     (591) 


The  quantity 


an 


=  K,  say,  is  called  the  constant  of  aber- 


k  4/1  - 
ration. 

Substituting  in  (584)  these  values  of  the  differential  co- 
efficients, we  therefore  have 


a'  —  a  =  —  K  sec  6[sin  O  sin  a  +  cos  O  cos  a  cos  w] 

sin  i"  sec2  8[(i  +  cos2  a>)  sin  za  cos  2©  —  2  cos  w  cos  za.  sin  2©] 

4 

K2 

—  Kf  sec  6  [sin  F  sin  a  -f-  cos  F  cos  a  cos  w]  -| sin  i"  sec2  8  sin  2a  sin2  w; 

6'  —  5  =  —  >c[sin  &  cos  a  sin  Q  —  (cos  w  sin  8  sin  a  —  sin  w  cos  8)  cos  0] 

—  Y  sin  i"  tan  8 1  [(i  +  cos  2  w)  cos  2  a  —  sin2  w]  cos  2  ©+2  cos  w  sin  2  ©  sin  20.  | 

—  «[sin  8  cos  a  sin  F  —  (cos  w  sin  8  sin  a  —  sin  w  cos  8)  cos  F] 
sin  i"  tan  8[(i  -|-  cos2  w)  —  sin2  w  cos  20.]. 


(592) 


§354-  REDUCTION   TO  APPARENT  PLACE.  609 

The  last  two  terms  in  each  are  constant,  or  are  only  suoject 
to  a  slow  secular  change  ;  they  will  therefore  be  combined 
with  the  mean  right  ascension  and  declination  of  the  star, 
and  will  require  no  further  consideration  in  this  connection, 
as  we  are  only  concerned  with  the  periodic  terms. 

The  most  commonly  received  value  of  the  constant  n  is 
that  of  Struve,  who  found  from  a  very  carefully  executed 
series  of  observations  at  the  observatory  of  Pulkova 
H  =  2o".445i.  (Recently  Nyre"n  finds  from  a  still  more  ex- 
haustive investigation  2o".492.)  For  1875.0  the  mean  value  of 
the  obliquity  of  the  ecliptic  is  co  =  23°  27'  19". 

Substituting  these  values  in  (592),  and  dropping  the  con- 
stant terms,  we  have  finally 

a'  —  a  =  —  20". 4451  sec  <5[sin  Q  sin  a  -f-  cos  O  cos  oc.  cos  (»] 

—  .0009330  sec'2  d  sin  20.  cos  20 
-j-        .0009295  sec2  d  cos  ice.  sin  2O; 

d'  —  d  =  —  20". 445 1  sin  d  cos  a  sin  Q  j>  (593) 

-}-  20  .4451  cos  O  [sin  d  sin  a  cos  GO —  cos  d  sin  GJ] 

—  .0004648  tan  d  sin  za  sin  20 

-{-      [.0000401  —  .0004665  cos  20]  tan  d  cos  20. 

Reduction  to  Apparent  Place. 

354.  We  have  now  deduced  the  essential  formulae  for  re- 
ducing a  star  from  mean  to  apparent  place  or  the  converse. 
The  place  as  given  in  the  star  catalogue  will  be  the  mean 
place  for  the  beginning  of  the  year  of  the  catalogue.  The 
reduction  of  this  place  to  the  mean  place  at  any  other  date 
has  been  explained  and  illustrated  with  sufficient  fulness, 
In  applying  the  formulae  as  we  have  done  we  obtain  the 
mean  place  for  the  beginning  of  the  year,  to  which  we  re- 
duce the  star's  co-ordinates.  If  now  we  wish  to  reduce  this 
mean  place  to  the  apparent  place  at  a  time  r  from  the  be- 
ginning of  the  year  (r  being  expressed  as  a  fraction  of  a  year), 
we  must  add  to  the  mean  right  ascension  and  declination  the 


6 10  PRACTICAL   ASTRONOMY.  §  354. 

precession  and  proper  motion  for  the  time  r,  as  given  by 
formulae  (565) ;  the  result  is  the  mean  place  at  time  r.  To 
this  mean  place  the  nutation  being  added  as  given  by  (579), 
we  have  the  true  place;  finally  adding  the  aberration  (593), 
we  have  the  required  apparent  right  ascension  and  declina- 
tion of  the  star. 

The  following  are  the  formulae  written  out  in  full,  omitting 
those  terms  in  the  nutation  and  aberration  which  are  ordina- 
rily inappreciable: 

a'  —  a  =  (nt  -f-  n  sin  a  tan  <5)  r  -\-  TJJ. 

—  (15". 8148  -j-  6". 8650  sin  «  tan  d)  sin  Q 

15  .8321        6  .8683 
-f- (       .1902-)-      .0825  sin  a  tan  d)  sin  2  Q 

—  (       .1872-!-      .0813  sin  «  tan  <5)  sin  2([ 

-\-  (       .0621  -)-      .0270  sin  a  tan  d)  sin  «[   —  F') 

—  (  i  .1642  -(-      .5054  sin  a  tan  d)  sin  2© 

-j-  (       .H73  +      -0509  sin  a  tan  S)  sin  (Q  —  F) 

—  (       .0195  -f-       .0085  sin  a  tan  £)  sin  (0  -f-  F) 

—  9  .2231  cos  a  tan  5  cos  Q  -)-  .0897  cos  a  tan  £  cos  2  Q 
9  .2240 

—  .0886  cos  a  tan  d  cos  2(£  —  -55O9  cos  a  tan  d  cos  20 
.0093  cos  a  tan  5  cos  (©  -\-  I~) 


20  .4451  cos  oa  sec  5  cos  a  cos  © 
20  .4451  sec  d  sin  or  sin  0  ; 


d'  —  d  =  rn  cos  a  -\-  T^' 

—  6".  8650  cos  a.  sin  Q  -|-  9". 2231  sin  a  cos  Q 
6  .8683  9  .2240 

-f-   .0825  cos  or  sin  2  Q  —   .0897  sin  a  cos  2  Q 

—  .0813  cos  a  sin  2 (£  -|-   .0886  sin  a  cos  2(C 
-|-   .0270  cos  a  sin  (C  —  F') 

—  .  5054  cos  a  sin  2 ©  -j-  ".5509  sin  a  cos  2© 
-j-   .0509  cos  a  sin  (©  — F) 

—  .0085  cos  a  sin  (0  -f-  F)  +  ".0093  sin  a  cos  (©  -}-  P) 
~  2o".445i  cos  GO  cos  ©  (tan  GO  cos  5  —  sin  a  sin  5) 

—  20  .4451  cos  a  sin  5  sin  0. 

The  values  of  the  constants  are  determined  for  1800.0. 
Where  the  change  is  appreciable  the  value  for  1900.0  is 
written  below. 


•  (594) 


§355-  REDUCTION   TO  APPARENT  PLACE.  6 1  I 

355.  The  formulas  as  written  above  are  complicated  and 
very  inconvenient  for  practical  application.  If  no  method 
could  be  devised  for  abridging  the  work,  star  reduction 
would  be  such  a  formidable  undertaking  that  but  little  prog- 
ress would  be  possible  in  this  direction.  The  method  in 
common  use,  however,  originally  proposed  by  Bessel,  re- 
duces the  labor  to  a  small  fraction  of  that  required  for  apply- 
ing the  formula  directly. 

It  will  be  observed  that  the  first  part  of  (a1  —  a]  consists 
of  a  number  of  terms  which  have  a  factor  of  the  general  form 
(m1  -j-  ri  sin  a  tan  tf),  the  constants  ni  and  ;/'  in  each  case 
having  nearly  tfte  same  ratio  to  each  other  as  m  to  n  in  the 
precession  formulae,  viz.,  2.3  approximately.  Therefore  let 

6.8650  =  ni\  15.8148  =  mi     -f  h\ 
.0825  =  ni'\  .1902  =  mi'    -f  /*'; 

.0813  =  ni"\  .1872  =  mi"  +  h"\ 

.0270  =  ni"'\  .0621  =  mi'"  +  h"'\  \  .     (595) 

.5054  =  niiv;  1.1642  =  mF  +  >^iv; 
.0509  =  niv;  .1173  =  m?    -f-  //v; 

.0085  =  ;«vi;  .0195  =  mi**  -f  h^. 

By  introducing  these  values  equations  (594)  may  be 
written 

a'  —  a.  -}-  Tfj.  -j-  \r  —  z'sin  Q-^-i'  sin  2Q—i"  sin  2<£-f  *""  sin  «[  —  F')— ziv  sin  20 
+  iv  sin  (Q  —  F)  —  zvl  sin  (Q  -f-  T)]  X  [m  +  «  sin  a  tan  5] 

—  [9". 2231  cos  Q  —  o". 0897 cos  2Q  -f-o".o886cos2(i;  -|- o". 5509  cos  20 

-j-  o".oo93  cos  (0  -f  Fj]  cos  a:  tan  d 

—  20". 4451  cos  GO  sec  8  cos  a  cos  0  —20". 4451  sec  d  sin  a  sin  O 

—  >$  sin  Q  -f  A' sin  2  Q  —  h"  sin  2([  -J-  ^'"  sin  (C  —  -^")  —  Alvsin  2© 

+  Avsin(0  —  T)  —  /^vlsin(0  -j-T); 
5'  =r  d  -j-  r/^'-f-  [r  —  »  sin  Q  +  i'  sin  2  Q  —  i"  sin  2(C  -|-  i'"  sin  (^  —  F') 

—  z'iv  sin  20  -f  zv  sin  (0  —  F)  —  i"  sin  (0  -f  T)]  X  n  cos  a 

-f  [9". 2231  cos  Q  —  o".o897  cos2&  -f  o". 08 86  cos  2(£  + -55C>9  cos  20 

+  0.0093  cos  (©  4-F)]  sin  a: 

—  20". 4451  cos  GO  cos  ©(tan  GO  cos  d  —  sin  a  sin  5) 

—  20". 4451  cos  a  sin  #sin  ©. 


6l2  PRACTICAL   ASTRONOMY.  §  355- 

It  will  be  observed  that  the  corrections  to  the  mean  values 
of  a  and  8  consist  of  terms  made  up  of  two  classes  of  factors, 
the  first  class  independent  of  the  star's  place  and  varying 
with  the  time,  the  other  class  depending  on  the  star's  place 
and  varying  so  slowly  that  they  may  be  regarded  as  constant 
for  a  considerable  time.  Writing  them  in  accordance  with 
Bessel's  original  notation, 

*A  =       T—  zsin  Q-fz"  sin  2Q  —  i"  sin  2  £-{-*""  sin((£  —  F')  —  *1v  sin  2Q 

+  *v  sin  (Q  -  T)  -  zvi  sin  (O  +  -T); 
B  =  —  9".  2231  cos  Q  +".0897  cos  2&  —  .0886  cos  2([  —  .55090032© 

—  .0093  cos  (O  +T); 
C  =  —  20".  4451  cos  oo  cos  O  ; 
D  =  —  20".  445  1  sin  O; 

E  —  —  h  sin  Q  -{-A'  sin  2  Q  —  h"  sin  2  £  -f  h'"  sin  (([  —  T')  —  fi"  sin  2  O 
V-  h"  sin  (O  -  r>  -  >fcvi  sin  (Q  -f  T); 

0  =  ^(fl*  -j~  n  sin  a  tan  <5);f         a   =  n  cos  a; 

£  =  Y&COS  a  tan  d;  b'  =  —  sin  a; 

c  —  ^  cos  a  sec  5;  <:'  =  tan  o>  cos  5—  sin  a  sin  5; 

</  =  ^  sin  a:  sec  5;  d'  —  cos  a  sin  5. 

Then  our  formulas  become 


a   =  --  ,       . 

Cc'  +Ddf.      \ 


A,  B,  C,  D,  E  being  the  same  for  all  stars  are  computed  in 
advance  for  every  day  throughout  the  year,  and  the  values 
given  in  the  nautical  almanac  and  the  similar  publications  of 
other  countries;  so  for  our  purposes  we  need  only  take  them 
from  these  sources. 

In  some  star  catalogues  a,  b,  c,  d  and  a',  b'  ,  c'  ,  d'  are  given 
in  connection  with  the  star's  place.  For  the  purposes  of  an 
accurate  reduction,  however,  these  become  obsolete  in  a  few 
years,  as  m,  ;*,  a,  d,  and  <*?  are  all  subject  to  slow  secular 

*  See  Art.  358. 

f  These  are  divided  by  15,  since  the  right  ascension  is  generally  given  in  time. 


§356. 


REDUCTION   TO  APPARENT  PLACE. 


613 


changes.     It  will  be  advisable  to  recompute  them  if  much 
time  has  elapsed. 

Example.     Required  the  apparent  place  of  a  Lyrae,  1884,  November  IO.  for 
upper  transit,  Washington. 

Mean  a  =  i8h  33m  o'.678  Mean  8  =  38°  40'  34".4O 


Then 

N.  A.  p.  284, 

Mean  place 


log  a  = 

0.3039 

".0179 

log  6  = 

i 

7.884a 

IL    —    +-                   ".2726 
T  =  0.863 

^m  —  3B.O724     )    by  formulae 
n  =  20".  05  34  i"        (549). 

log  c  =  8.0884            log  d  —  8.926gn 

log  A  = 

9.9602             logB  = 

0.9619 

logC 

=  1.0894 

\ogD  = 

1.1883 

log  a'  = 

0.4592 

log  b'  = 

9-9955 

logc' 

=  9.9809 

log  d'  = 

8.9528 

log  A  a  — 

0,2641 

log  Bb  = 

8.8461 

log  Cc  =  9.1778 

logDd  = 

O.II52n 

log  A  a'  = 

0.4194 

log  BV  = 

0-9574 

log  Cc' 

=  1.0703 

log  Dd'  = 

O.I4II 

a 

- 

i8h 

33m  o8.678 

d 

— 

38° 

40'  34"- 

40 

Aa 

— 

i  .837 

Aa 

= 

2    . 

63 

Bb 

— 

.070 

Bb' 

— 

9  • 

07 

Cc 

— 

.150 

Cc' 

= 

ii  . 

76 

Dd 

= 

— 

i  -304 

Dd' 

= 

i  . 

38 

E 

= 

.001 

rp 

= 

.016 

TV' 

= 

23 

ace 

a' 

: 

i8h 

33m  I8.448 

8 

=r 

38° 

4°'  59"- 

47 

Apparent  place 

356.  The  above  form  of  reduction  is  most  convenient  when  a  considerable 
number  of  apparent  places  is  required,  or  when  the  star  catalogue  gives  reliable 
values  of  the  constants  a,  b,  C,  d,  etc.  If  these  quantities  are  not  given  and 
only  one  or  two  apparent  places  are  required,  a  different  form  may  be  given  to 
equations  (597)  which  will  be  more  convenient.  This  transformation,  also  due 
to  Bessel,  is  as  follows: 


Write 


Then  we  have 


f=mA+E; 

g  cos  G  =  nA  ; 
g  sin  G  —  B\ 


i  =  C  tan  GO  ; 
h  cos  H  =  D\ 
hsinff=  C. 


a'  —  a  4-  TH  -f/ 

8'  =  5  +  Tfi'  4-  i  cos  S  -f  g  cos  (G  +  a) 


-f  g  sin  (G  -f  «)  tan  d  -\-  h  sin  (H  +  a)  sec  d;  )      ,    ~ 
-f-  h  cos  (ff+  a)  sin  8.  ) 


The  values  of  r,  f,  G,  H,  log  g,  log  h,  and  log  i  are  also  given  in  the  epheme- 
ris  for  every  day  of  the  year. 


6  14                                  PRACTICAL   ASTRONOMY.  §  357. 

As  an  example,  let  these  formulae  be  applied  to  determine  the  apparent  place 
of  a  Lyrae  on  the  date  given  above. 

We  have                           a  =  i8h  33™.  o  £  =  38°  40'.  6 

page  291  of  ephemeris,      G  =     I   46   .3  *G-{-a  =  2oh  19™.  3 

H  =    2   34   .2  *H-{-a  =  21      7   .2 

log  TV  =  8.8239  log  If  =  8.8239 

page  291  of  ephemeris,      log^-  —  1.3109  log  h  =  1.2952 

*sin  (G  +  a)  =  g.gi42n  *sin  (H  +  a)  —  9.8373 

tan  d  =  9.9033  sec  6  =     .1075 

log  (£)  =  9-9523«  log  (/&)  =    .0639* 

log£-  —  1.3109  log  h  =  1.2952 

cos  (G  -\-  a)  =  9.7570  cos  (H  -\-  a)  =  9.8610 


log  U')  =  1.0679  sin  s  =  9-7958 

page  291  of  ephemeris,      log  i  =  0.7273  log  (^')  =  0.9520 
cos  d  =  9.8925 

log  (i)  =  0.6198 

a  —  i8h  33m  o8.678  d  =  38°  40'  34".  40 

/=                2.804  (£-')  =               ii  .70 

<*)=-              -895  (*')  =                 8.95 

(A)=-            1.158  (0=                 4-17 

rfj.  =                    .016  T)JL'  =                      .23 


«'  =  i8h  33m  is.445  <5'  =  38°  40'  59".  45 

357-  -M^-     Certain  of  the  small  terms  which  have  been  neglected   in  the. 
preceding  formulae  will  sometimes  be  appreciable  for  stars  near  the  pole  where 
great  accuracy  is  required. 

1st.  The  Precession  for  Time  r.  We  have  only  used  the  term  depending  on 
the  first  power  of  r.  The  values  of  the  second  differential  coefficients  are 
given  by  equations  (565).  The  numerical  values  being  substituted,  the  only 
terms  which  can  be  appreciable  are 

A(cd  —  or)  =  +  8-000  003  r8  sin  a-  tan  d  —  O8.ooo  I4gr2  cos  a  tan  d  j 

—   .oooo6sr2  sin  ia  tan8  §;  >     (599) 

,4(8'  _  5)  =  _|_  .000  975  ra  sin3  a  tan  d.  J 

2d.  In  the  formulae  for  aberration  (593),  rigorously  a,  d,  ©,  and  ft?  are  not 
the  mean  values  of  these  quantities  as  there  assumed,  but  the  true  values.  They 

*  A  table  giving'  logarithmic  sines  and  cosines  with  the  argument  expressed  in  time  is  con- 
venient. If  this  is  not  available,  (G  -f-  <*•)  and  (H  '-}-  o)  must  be  reduced  to  arc. 


§357- 


SEDUCTION  TO  APPARENT  PLACE. 


6l5 


should  therefore  be  corrected  for  nutation.  The  necessary  corrections  to 
(a!  —  a)  and  (d'  —  d)  as  given  by  (593)  may  be  determined  by  differential 
formulae. 

Since  (a1  —  a)  =  f(a,  d,  Q,  oo),  and  similarly  for  (d'  —  d), 


da 


do 


da. 


dw 


doo 


(600) 


Where  Aa,  Ad,  etc.,  represent  the  corrections  for  nutation  given  by  (572)  and 

(579). 

Practically  the  terms  in  AQ  and  AGO  will  never  be  appreciable,  and  of  the 
values  of  Aa  and  Ad  we  need  only  retain  the  following  terms: 


Aa  =  —  [6".  865  sin  or  sin  Q  -{-  9".  2235  cos  a  cos  Q]  tan  d; 


~. 


Ad  =  —  6". 865  cos  a  sin  Q  -|-  9". 2235  sin  a  cos  Q. 
Differentiating  (593)  with  respect  to  a  and  6",  neglecting  the  smaller  terms, 

d(a'  -  a) 

=  —  20  .4451  sec  o[cos  a  sm  O  —  sin  a  cos  O  cos  flftj; 

d(a'  -  a] 

-j-ft —  =  —  20  .4451  sec  o  tan  o[sin  a  sin  O  +  cos  a  cos  0  cos  GO]; 

— -j =        2o".445i  [sin  d  sin  a  sin  ©  -J-  sin  S  cos  a  cos  ©  cos  GO]; 

d(d'  -  d) 

jft —    =  —  20  .4451  cos  0  cos  a:  sin  © 

+  20". 4451  cos  ©  [cos  S  sin  a  cos  GO  -{-  sin  d  sin  GO]. 

Substituting  in  (600)  and  retaining  only  terms  multiplied  by  tan  d  or  sec  I 
we  find 


A(a'-a)=  -  20//'445Isin  i"  tan  5  sec  8 


sin  i"  sin  8  tan  8 


— (6".865-f  9"  2235  cos  w)  sin  2a  cos  ( Q  -|-  Q  ) ; 

•K6  .865  cos  w-f  9".2235)  cos  20.  sin(  Q  -f-  Q  ) ; 

+(6  .865— 9". 2235  cos  w)  sin  2acos(Q  —  Q); 

—(6  .865 cos  w— 9". 2235)  cos  20.  sin(Q  —  Q)j 

—  (6  .865+9"  2235  cosw)  cos 2a cos( 0 -j-  Q); 

—(6  .865  cos  w-|-9".2235)  sin  2a  sin(  0  -{-  Q  }', 

-f-(6  .865— 9". 2235  cosw)  cos  2acos(O  —  Q); 

-f(6  .865  cos  w-9". 2235)  sin  2asin(0  —  Q); 

+(6  .865-9//.2235  COS  w)  cos(Q-(-,Q); 

-(6  .865+9".2235  COS  a>)  cos  (Q  —  Q  ). 


(602) 


6i6 


PRACTICAL  ASTRONOMY. 


§353, 


These  expressions  reduce  to  the  following: 

—  ".ooo  05065  sin  2or  cos  (O  +  Q) 
-a=      +  -ooo  05129  cos  2or  sin  (0+Q) 

—  .000  00527  sin  2tr  cos  (0  —  £) 
-f-   .000  00966  cos  20:  sin  (0  —  Q) 


4(3' -3)=* 


f  —  ".ooo  3799  cos  2a  cos  (0  -f  Q) 
.000  3847  sin  2a  sin  (©  -f-  £) 

.000  0395  COS  2(X  COS  (©    —    Q) 

.000  0725  sin  2or  sin  (0  —  Q) 
.000  0391  cos  (0  -j-  Q) 
.000  3799  cos  (0  -  Q) 


sin      tan 


(603 


3d.  In  a  few  cases  of  double  stars  the  maan  place  oi  ".he  star  requires  a  cor 
rection  for  orbital  motion.  The  corrections  to  the  right  ascension  and  declina 
tion  will  have  the  form 

Aa  =  a  -f-  bt  -f  k  sin  (n  -f-  H): 
AS  —  a  -f  £'/  -f-  £'  sin  (n  -f-  H'); 

the  quantities  entering  into  the  formulae  depending  on  the  elements  of  the  star?; 
orbit. 

358.  The  foregoing-  comprises  all  that  is  necessary  for  re- 
ducing stars  from  mean  to  apparent  place,  or  from  apparent 
to  mean  place.  In  the  latter  case  the  corrections  will  be 
applied  with  the  opposite  signs  to  those  given  by  formulae 
(597)  or  (598).  Since  1834  the  factors  A,  B,  C,  D  have  been 
published  by  the  British  Nautical  Almanac,  and  in  the  Ameri- 
can Ephemeris  since  its  first  publication,  1855.  In  the  Brit- 
ish Almanac  and  previous  to  1865  in  the  American  Ephemeris 
the  notation  is  not  Bessel's  which  we  have  given,  but  that  of 
Baily,  viz.,  A  is  interchanged  with  C,  and  B  with  D*  Particu- 

*  This  unnecessary  and  confusing  change  of  notation  was  introduced  by 
Baily  for  no  better  reason  than  the  following:  "I  have  thought  it  desirable 
that  we  should  as  much  as  possible  make  them  serve  the  purpose  of  an  artificial 
memory.  It  is  on  this  account  that  I  have  made  AB  represent  the  quantity  by 
which  ABerration  is  determined;  Cthe  quantity  by  which  preCession  is  de- 
termined ;  and  D  the  quantity  by  which  the  Deviation,  or  (as  it  is  now  more 
generally  called)  the  nutation,  is  determined." — British  Association  Catalogue, 
p.  34,  note. 


§359-  THE  FICTITIOUS    YEAR.  617 

lar  attention  must  therefore  be  given  to  the  notation,  other- 
wise errors  will  be  very  likely  to  occur.  Since  1865  the 
notation  of  Bessel  has  been  employed  in  the  American 
Ephemeris. 

For  any  date  from  1750  to  1850  the  logarithms  of  At  B, 
C,  D  may  be  taken  from  Bessel's  Tabula  Regiomontancs. 
Bessel's  constants  are  employed  and  the  smaller  terms  are 
neglected ;  they  will,  however,  give  all  necessary  precision 
in  the  few  cases  where  it  will  be  found  necessary  to  employ 
them.  A  convenient  table  by  Hubbard  for  correcting  them 
so  as  to  make  the  values  conform  to  the  constants  of  Struve 
and  Peters  will  be  found  in  Gould's  Astronomical  Journal, 
vol.  iv.  p.  142.  Bessel's  tables  are  computed  for  every  tenth 
day  of  the  fictitious  year.  Their  employment  involves  a  sub- 
ject the  consideration  of  which  we  have  not  found  necessary 
heretofore,  viz., 


The  Fictitious  Year. 

359.  We  have  heretofore  spoken  of  the  year  without  speci- 
fying very  definitely  which  of  the  various  periods  called  a 
year  was  to  be  understood.  The  common  year  is  not  well 
adapted  to  the  requirements  of  astronomy,  since  the  length 
is  not  the  same  in  all  cases,  each  fourth  year  containing  one 
more  day  than  the  other  three.  The  Julian  year  of  365 J 
days  is  better,  but  its  length  does  not  exactly  correspond  to 
the  movements  of  the  earth  in  its  orbit. 

In  the  reduction  of  star  places  Bessel  obviates  the  difficul- 
ties which  would  follow  from  the  employment  of  either  of 
the  above  periods  by  employing  a  fictitious  year  to  begin  at 
the  instant  when  the  longitude  of  the  mean  sun  is  280°.  This 
instant  will  of  course  not  coincide  with  the  transit  of  the  siyi- 
over  the  meridian  of  Greenwich  or  Washington,  but  from 


PRACTICAL   ASTRONOMY.  §360. 

the  known  mean  motion  of  the  sun  the  Greenwich  or  Wash- 
ington time  may  be  found  at  which  the  mean  longitude  is 
280°,  and  consequently  the  meridian  over  which  the  sun  is 
passing  at  this  instant.  This  is  sometimes  called  the  normal 
meridian,  and  may  then  be  employed  as  the  prime  meridian 
from  which  to  reckon  longitudes  throughout  the  year  pre- 
cisely as  the  meridians  of  Greenwich  and  Washington  are 
used.  Since  the  sun's  mean  right  ascension  equals  the  mean 
longitude,  the  sidereal  time  at  this  meridian  corresponding 
to  the  beginning  of  the  year  will  be  i8h  40™  (=  280°).  If  then 
we  imagine  a  point  on  the  celestial  equator  whose  right 
ascension  is  i8h  40™,  the  sidereal  day  throughout  the  fictitious 
year  may  be  regarded  as  beginning  at  the  instant  when  this 
point  crosses  the  meridian,  just  as  in  the  common  method 
the  sidereal  day  begins  when  the  vernal  equinox  crosses  the 
meridian.  By  adopting  this  device  a  uniformity  and 
simplicity  is  introduced  into  those  quantities  which  are 
functions  of  r.  This  is  also  the  date  to  which  the  mean 
places  of  stars  are  reduced  in  the  star  catalogues.  When 
the  elements  of  reduction  are  taken  from  the  Nautical 
Almanac  or  American  Ephemeris  no  attention  need  be  given 
to  this  matter,  as  it  is  already  provided  for. 

Bessel  calls  the  instant  when  the  sun's  mean  longitude 
equals  280°  Jan.  o.o  of  the  fictitious  year.  This  corresponds 
to  Dec.  31.0  of  the  usual  method  of  reckoning;  that  is,  accord- 
ing to  Bessel's  method  Jan.  i,  2,  3,  etc.,  indicate  i,  2,  3,  etc., 
days  from  the  beginning  of  the  year,  while  in  the  common 
method  the  beginning  of  the  ist,  2d,  etc.,  days  is  understood. 

We  shall  now  show  the  relation  between  the  beginning  of 
the  fictitious  and  common  years,  afterwards  returning  to  the 
Tabula  Regiomontance. 

360.  During  one  complete  century  the  period  of  the  com- 
mon year  is  the  same  as  that  of  the  Julian  year.  Suppose 
now  for  the  moment  that  at  1800.0  the  fictitious  year  began 


§  360.  THE  FICTITIOUS    YEAR.  619 

with  the  date  Jan.  o.o  of  the  common  year,  and  that  the 
length  of  the  tropical  year  coincided  with  that  of  the  Julian. 
Then  for  any  other  date  1800  +  /  we  should  have 

Beginning  of  year  =  Jan.  o.o  +  i/>     •     •     (604) 

where /is  the  remainder  after  dividing  the  number  of  the 
year  by  4.  In  case  of  a  leap-year,  where  the  number  of  the 
year  is  exactly  divisible  by  4,/  must  be  made  equal  to  4, 
since  the  intercalary  day  is  not  introduced  until  the  end  of 
February. 

If  we  choose,  in  accordance  with  Bessel,  as  our  prime 
meridian  that  of  Paris,  the  above  formula  involves  two  erro- 
neous assumptions :  first,  the  beginning  of  the  year  from 
which  we  reckon  will  not  coincide  with  Jan.  o.o;  and  second, 
the  length  of  the  tropical  year  is  not  that  of  the  Julian.  We 
shall  use  the  constants  of  Bessel  in  order  to  have  our  results 
those  of  the  Tabula  Rcgiomontance. 

For  mean  noon  at  Paris,  viz.,  1800,  Jan.  o.o,  Bessel  finds 
for  the  sun's  mean  longitude 

279°  54'  i".3<>, 

and  for  the  mean  daily  motion  of  the  sun  in  longitude 
3548//-33°2  +  ".ooo  ooo  6902*,* 

where  t  —  number  of  years  elapsed  since  1800. 

For  the  meridian  of  Paris  we  must  add  to  (604)  the  time 
required  for  the  sun  to  move  358//.64,'viz.,  0.10107289  day. 

It  remains  to  correct  (604)  for  the  difference  between  the 

*  It  will  be  observed  from  the  expression  for  the  mean  daily  motion  that  the 
length  of  the  year  is  not  constant ;  the  variation,  however,  amounts  only  to 
°8-595  per  century. 


620  PRACTICAL   ASTRONOMY.  §  361. 

Julian  and  tropical  years.     The  tropical  motion  of  the  sun  in 
one  Julian  year  is,  according  to  Bessel, 


360°  oo'  27 

Therefore  the  mean  tropical  motion  in  /  years  will  be 
[360°  oo'  27".6o5844]/  +  o".oooi22i8A 


The  time  required  for  the  mean  sun  to  pass  over  the  dis- 
tance 27".6o5844/,  expressed  as  a  fraction  of  a  day,  will  be 
.0077799535/  +  o.ooo  ooo  034433/2.  Therefore  the  complete 
formula  for  the  Paris  mean  time  of  the  beginning  of  any  ficti- 
tious year  will  be 

Jan.  o.o  -{-  o.dioio7289  —  o.doo77799535/  —  0.000000034433^  -f-  i/.  (605) 

To  reduce  any  mean  solar  date  at  Paris  to  the  date  of  the 
fictitious  year  the  above  quantity  must  be  subtracted. 
Therefore  let 

k  —  —  o.  10107289  -f  o  0077799535/  -f  o.ooo  ooo  034433^  —  i/ 

k  is  then  the  longitude  east  from  Paris  of  the  meridian  where 
the  fictitious  year  begins,  or  of  the  normal  meridian. 

Let  d  =  the  longitude  west  of  Paris  of  any  meridian,  ex- 
pressed as  a  fraction  of  a  day. 

Then  the  reduction  which  must  be  applied  to  any  mean  solar 
date  at  this  meridian  to  reduce  it  to  the  normal  meridian  is 
k  +  d. 

361.  Let  us  now  return  to  the  Tabula  Regiomontance.  The 
logarithms  of  A,  B,  C,  D,  T,  and  the  quantity  E  are  there 
given  for  every  tenth  day  of  the  fictitious  year  from  1750  to 
1850;  the  intervals  being  sidereal  instead  of  mean  solar  days, 
an  arrangement  which  is  a  little  more  convenient  in  star  re- 

*  This  quantity  divided  by  365.25  is  the  mean  daily  motion  already  given. 


§  361.  THE    TABULA  REGIOMONTAN^E.  621 

duction,  for  the  reason  that,  the  star  being  generally  observed 
on  the  meridian,  its  right  ascension  is  at  once  the  sidereal 
time  of  observation.  In  order  to  apply  the  tables  we  must 
first  convert  this  sidereal  time  to  the  corresponding  sidereal 
time  at  the  normal  meridian. 

It  will  be  remembered  that  the  sidereal  day  of  the  ficti- 
tious year  at  any  meridian  begins  at  i8h  4Om  sidereal  time; 
therefore  at  this  meridian  itself  the  tables  are  applicable  for 
this  instant  of  local  time.  For  any  other  meridian  at  the 
instant  i8h  4Om  local  sidereal  time  the  argument  of  the  tables 
will  be  k  -f-  d. 

At  any  other  sidereal  time  g  at  this  last  meridian  the  argu- 
ment will  be 

,    ,     ,  ,  g 
k  +  d~ 

which  must  be  less  than  unity  and  positive.  Or  we  may 
write 

5h 


as  the  quantity  to  be  added  to   b  -\-  d,  omitting  one  whole 
day  when£-  +  5h  2Om  =  or  >  24**. 

If,  as  before  assumed,  we  regard  the  sidereal  day  of  the 
fictitious  year  as  beginning  when  the  right  ascension  of  the 
meridian  is  i8h  40™,  then  as  long  as  the  right  ascension  of  the 
sun  is  less  than  this  quantity  it  will  cross  the  meridian  before 
the  point  on  the  equator  having  this  right  ascension,  and  the 
day  of  the  fictitious  year  will  be  the  same  as  the  common 
date.  When  the  sun's  right  ascension  is  equal  to  i8h  40™ 
(the  sun  being  on  the  meridian)  the  two  days  begin  together, 
and  when  it  is  greater  than  i8h  40™  the  sidereal  day  of  the 
fictitious  year  begins  before  the  common  day,  and  therefore 
one  da}'  must  be  added  to  the  common  reckoning  for  the 


622  PRACTICAL   ASTRONOMY.  §  361. 

date  of  the  fictitious  year.  Therefore  the  argument  of  the 
table  will  be 

k  +  d  +  gf  +  *, 

in  which  /  =  ofrom  beginning  of  the  year  to  where  the  right 
ascension  of  the  mean  sun  equals  the  sidereal  time,  after 
which  i  =  i. 

The  Tabulce  Regiomontance  then  give  the  following 
quantities: 

Table  I  gives  k  for  the  longitude  of  Paris  expressed  in 
hours,  minutes,  and  seconds,  and  also  as  a  fraction  of  a  day, 
for  every  year  from  1750  to  1849. 

Table  II  gives  d,  the  west  longitude  from  Paris  of  a  num- 
ber of  the  principal  cities  of  Europe.  (Better  values  can, 
however,  be  found  in  the  ephemeris.) 

£  ~\~  5h  2Om 

Auxiliary  table,  p.  16,  gives  g'  —  -  — . 

24 

Table  VIII,  pp.  17-116 inclusive,  gives  log  A,  log  B,  log  C, 
log  />,  log  r,  and  E. 

For  C  and  D  table  IX  may  be  employed.  It  requires  no 
special  explanation  here. 

Example.  Required  the  logarithms  of  A,  B,  C,  D,  r,  for 
1825,  July  id  ioh,  Greenwich  sidereal  time. 

Table  I  for  1825,  k  =  —-.157 

Table  II  for  1825,  d  =  +  .007 

Page  16,  gr  =  +  .639 

i  —       .000 


Argument  =  July  1.489 

Page  92,  table  VIII,       log  A  =  9.9224 

log  B  =  0.3026          E  =  +  ".05 
log   T  =  9.6975 
table  IX,      log  C  =    .4817 
logZ>  =  i.30o6n 


§  362.  MEAN  SOLAR  AND   SIDEREAL    TIME,  623 

The  quantities  have  been  interpolated  directly  from  the 
tables;  log  C  and  log  D  are  given  more  accurately  by  table 
IX.  If  thought  desirable,  the  interpolation  may  be  carried 
out  to  second  differences,  but  this  will  not  often  be  necessary. 

As  an  example  of  a  case  where  i  =  i  let  it  be  required  to 
find  the  above  quantities  for  1825,  Dec.  id  ioh,  Greenwich 
sidereal  time. 

As  before,  k  —  --   .157 

d  =  +  .007 

Table  VI,  right  ascension  of  g'  =        .639 

Mean  sun  Dec.  i  is  i6h  40™,  therefore  i  =       i.ooo 


Argument  =  Dec.  2.489 

With  this  argument  we  find 

log  A  —  0.0867  ;         log  B  =    .4976 ;         log  C  —  .7599 ; 
log/>  =  1.2772  ;         log  r  =  9.9631  ;  E  =  +  .05. 

Various  forms  of  tables  for  star  reductions  have  been  pro- 
posed and  employed.  Some  of  these  are  very  useful  for 
special  purposes,  but  it  is  not  necessary  to  enter  into  the 
details  of  their  construction  in  this  connection. 

362.  Conversion  of  Mean  Solar  into  Sidereal  Time  and  the  con- 
verse. The  solution  of  this  problem  for  any  date  after  the 
British  and  American  Nautical  Almanacs  became  available 
in  their  present  form  has  been  treated  with  all  necessary  ful- 
ness in  Articles  94  and  95.  For  earlier  dates  other  methods 
must  be  used.  The  Tabula  Regiomontana  gives  the  data 
necessary  for  solving  the  problem  for  any  date  between  1750 
and  1850. 

We  have  shown  in  Art.  94  that  the  mean  time  at  any 
meridian  is  equal  to  the  true  hour-angle  of  the  second  mean 
sun,  which  moves  uniformly  in  the  equator,  and  whose  mean 


624  PRACTICAL   ASTRONOMY.  §  362. 

right  ascension  is  equal  to  the  mean  longitude  of  the  first 
mean  sun,  which  moves  in  the  ecliptic. 

Also,  the  sidereal  time  is  equal  to  the  hour-angle  of  the  true 
equinox.  Therefore  in  our  formula 

®=a  O  +  r.   ......     (199) 

a  O  must  be  understood  to  mean  the  true  right  ascension  of 
the  second  mean  sun.  This  equals  the  mean  right  ascension 
plus  the  nutation  of  the  vernal  equinox  in  right  ascension. 
The  latter  is  found  from  the  general  equations  (579),  by 
making  a  =  o,  d  —  o  to  be  ^/A  cos  &?,  and  is  given  in  the 
ephemeris  as  the  "equation  of  the  equinoxes  in  right  ascen-. 
sion."  It  is  included  in  the  sidereal  time  of  mean  noon 
given  bv  the  ephemeris.  When  the  ephemeris  is  available 
it  will  therefore  require  no  further  notice. 

Table  VI  of  the  Tabula  Regiomontance  gives  the  right  ascen- 
sion of  the  second  mean  sun  corrected  for  the  solar  nutation 
of  the  equinox  for  every  mean  noon  at  the  fictitious  meridian. 
The  fictitious  year  always  begins  with  the  same  right  ascen- 
sion of  the  mean  sun,  therefore  this  table  is  available  for 
every  year.  The  number  taken  from  this  table  for  any  date, 
which  must  be  the  date  at  the  normal  meridian,  is  then  cor- 
rected for  lunar,  nutation  in  right  ascension,  which  is  given 
by  table  IV.  The  result  is  the  sidereal  time  of  mean  noon, 
F0,  at  the  normal  meridian,  which  may  be  used  in  precisely 
the  same  way  as  the  sidereal  time  of  mean  noon  at  Washing- 
ton. (See  Articles  94  and  95.)  Or  writing  the  formulae 
out  in  full, 

0=  T  +  table  VI  +  table  IV  +  (T+k  +  d)(n'-  i);(£o6) 


or   r=  F0+  (£  +  </)(;/-  i)  =  VI  +  IV  +  (£  +  ^)  OK-  i), 

e=  r+  r+  T(v-  i). 


§  362.  MEAN  SOLAR  AND   SIDEREAL    TIME.  62$ 

And  for  converting  sidereal  into  mean  solar  time, 

T=9-  r-(9-  v)(i  -),);.  .  .   (607) 

The  notation  being  that  of  Articles  94  and  95. 

Example.      Given   1825,  July   id  7h  25"',  Greenwich  mean 
soiar  time.     Required  the  corresponding  sidereal  time. 

By  the  first  of  formulae  (606),  T  —  /h  25m    os.ooo 

Table  VI  =  6  37    33  .099 
Table  IV  =  i  .015 

(T+  £  +  </)(/*-  0>  Table  vn  =  37  -606 

T  —      7h  25m    os.ooo 

Table    I,  £  =  —  3   45    26.1  0  =  i4h  3m  iis.72 

Table  II,  d=+        9    21.6 


Example  2.     Given  1825,  July  id  I4h  3™  ii8.72,  Greenwich 
sidereal  time.     Required  the  corresponding  mean  solar  time. 

Table  VI  =    6h  37™  33S.O99 

(k  +  d)  =  -  3b  36m  4S.5  Table  IV  =  i  .015 

(k  +  d)(p-  i),  Table  VII  ==  -  35  .495 

V=    6h 


©  —  V  =    7h  26m  i3s.ioi 
Table  VII  =  -       i    13.101 

T  =      h  2m    os.o 


TABLES. 


Table   I   gives  values  of   the  function   —-=.  f'e  *dt   for 

r    TL^s  9 

values  of  t  from  o  to  oo  . 

Table  II  A  gives  the  refraction  corresponding  to  different 
altitudes  for  a  mean  state  of  the  atmosphere,  viz.,  barometer 
30  inches,  thermometer  50°.  For  any  other  readings  of  the 
barometer  and  thermometer  the  factors  by  which  the  mean 
refraction  must  be  multiplied  are  taken  from  tables  II  B, 
II  C,  and  II  D.  (See  Art.  86.) 

Table  III  A,  B,  C,  and  D  are  Bessel's  refraction  tables. 
These  will  be  employed  when  extreme  precision  is  required. 
When  the  altitude  is  less  than  5°  no  table  will  give  reliable 
values  for  the  refraction,  but  it  may  be  found  approximately 
by  the  supplementary  table  following  III  A.  (See  Art.  86.) 

Table  IV  is  intended  for  use  in  connection  with  the  refrac- 
tion table  when  the  barometer  is  graduated  according  to  the 
metric  system. 

Tables  V  or  VI  may  be  used  when  the  thermometer  is  not 
graduated  according  to  Fahrenheit's  scale. 

Table  VII  requires  no  explanation. 

Table  VIII  A  and  B  give  values  of  m,  log  m,  n,  and  log  n, 
where 

2  sin2  \t     •  2  sin4  %t 

m  =    sin  !//  '>        n  =    Sin  jf,  •        (See  Art.  146.) 

Table  VIII  C  gives  the  factor  to  employ  in  reducing  cir- 
cummeridian  altitudes  when  the  chronometer  has  an  appre- 

/         i  *      V 
ciable  rate,  viz.,  k  =  (  - \.     (See  Art.  152.) 

86400 


TABLE  I. 


627 


=   —   /    e         dt. 


t 

At} 

t 

M 

t 

At) 

t 

At) 

.00 

.000000 

•5° 

.520500 

I.OO 

.842701 

1.50 

.966105 

.01 

.011283 

•51 

.529244 

1.  01 

.846810 

51 

.967277 

.02 

.022565 

•52 

•537899 

1.  02 

.850838 

52 

.968414 

•°3 

.033841 

•53 

.546464 

I-Q.3 

.854784 

53 

'  .969516 

.04 

.045111 

•54 

•554939 

1.04 

.858650  ' 

54 

.970586 

•05 

.056372 

•55 

•563323 

1.05 

.862436 

55 

.971623 

.06 

.067622 

•S6 

.571616 

i.  06 

.866144 

.56 

.972628 

.07 

.078858 

•57 

•579817 

1.07 

.869773 

57 

•973603 

.08 

.090078 

•58 

•587923 

1.08 

.873326 

•58 

•974547 

.09 

.101281 

•59 

•595937 

1.09 

.876803 

•59 

.975462 

.10 

.112463 

.60 

.603856 

1.  10 

.880205 

.60 

.976348 

.11 

.123623 

.61 

.611681 

I.  II 

•883533 

.6! 

.977207 

.12 

.134754 

.62 

.619412 

1.  12 

.886788 

.62 

•978038 

•*3 

.145867 

.6* 

.627046 

«.*a 

.889971 

•63 

.978843 

.14 

.156947 

.64 

.634586 

1.14 

.893082 

.64 

.979622 

.15 

.167996 

•65 

.642029 

*«*5 

.896124 

•65 

•980376 

.16 

.179012 

.66 

•649377 

1.16 

.899096 

.66 

.981105 

-1? 

.189992 

.67 

.656628 

1.17 

J|b2COO 

.67 

.981810 

.18 

.200936 

.68 

.663782 

1.18 

^04837 

.68 

.982493 

.19 

.211840 

.69 

.670840 

1.19 

.907608 

.69 

•983153 

.20 

.222703 

.70 

.67780! 

1.20 

.910314 

•70 

.983791 

.21 

.233522 

•71 

.684666 

1.  21 

.912956 

•71 

.984407 

.22 

.244296 

•72 

•69143.1 

1.22 

•9I5534 

72 

.985003 

•23 

.255023 

•73 

.698104 

1.23 

.918050 

•73 

•985578 

.24 

.265700 

•74 

.704678 

1.24 

•920505 

•74 

.986135 

•25 
.26 

.276326 

.286900 

•75 
.76 

.711156 
•7T7537 

1:3 

.922900 
.925236 

•75 
76 

.986672 
.987190 

.27 

.297418 

•77 

.723822 

1.27 

•9275M 

•77 

.987691 

.28 

.307880 

•78 

.730010 

1.28 

.929734 

.78 

.988174 

.29 

.318284 

•79 

.736104 

1.29 

.931899 

•79 

.988641 

•30 

.328627 

.80 

.742101 

1.30 

.934008 

.80 

.989090 

•3« 

.338908 

.81 

.748003 

I-3I 

.936063 

.81 

•989525 

•32 

.349126 

.82 

•7538ii 

1.32 

.938065 

.82 

.989943 

•33 

•359279 

•83 

•759524 

L33 

.940015 

•83 

•990347 

•34 

•369365 

.84 

•765143 

1-34 

.941914 

.84 

.990736 

9 

.379382 
•38933° 

•85 
.86 

.770668 
.776100 

'•35 
1.36 

.943762 
•945562 

.86 
.88 

•991473 
.992156 

•37 

.399206 

.87 

.781440 

x-37 

•947313 

.90 

.992790 

•38 

.409010 

.88 

.786687 

r.38 

.949016 

.92 

•993378 

•39 

.418739 

.89 

.791843 

1-39 

•950673 

;   -94 

•993923 

.40 

.428392 

.00 

.796908 

1.40 

•952285 

!   .96 

•994426 

.41 

•437969 

.91 

.80.883 

1.41 

•953852 

.98 

.994892 

.42 

.447468 

.92 

.806768 

1.42 

•95S376 

1   '° 

•995323 

•43 

.456887 

•93 

.811564 

*'43 

•956857 

I 

.997021 

•44 

.466225 

•94 

.816271 

1.44 

.958296 

2.2 

.998137 

•45 

•475482 

•95 

.820891 

i-45 

•959695 

2-3 

.998857 

.46 

.484655 

.96 

•  8254:>4 

i  46 

.961054 

2.4 

.999312 

•47 
.48 

•493745 
.502750 

•97 
.98 

.829870 
.834232 

'•47 
1.48 

.96,373 
•963654 

2-5 

3-o 

•999593 
.999978 

•49 

.511668 

•99 

.838508 

1.49 

.964898 

3-5 

•999999 

.50 

.520500 

1.  00 

.042701 

1.50 

.966105 

oo 

1.  000000 

628 


TABLE  II  A. 


Barometer  30  inches. 


MEAN  REFRA  CTION. 

Fahrenheit's  Thermometer  50° 


Apparent  1 
Altitude.  | 

Mean 
Refraction. 

1 

Mean 
Refraction. 

c  w 

SI 

Mean 
Refraction. 

II 
II 

Mean 
Refraction. 

4->    . 

C  OJ 

SI 

fi<3 

Mean 
Refraction. 

C  V 

SI 

Mean 
Refraction. 

C  4) 

SI 

a— 

Mean 
Refraction. 

o°3o' 

29*19" 

8°35' 

6'  8".S 

i2.°35' 

4  *5  -3 

I9°io' 

2'46".I 

27°  io' 

i'53".i 

42°20' 

'  3  '-9 

79°oo' 

o'u".3 

I   O 

2438 

8  40 

6  5  .2 

12  40 

4  13  -6 

19  20 

2  44  .6 

27  20 

i  52  .3 

42  40 

3  -2 

80  o 

10  .3 

2   0 

1819 

8  45 

6  2  .O 

12  45 

4  12  .0 

19  30 

2  43  -i 

27  30 

43  o 

2  .4 

81  o 

9  -2 

3  o 

1422 

8  50  5  58  -8 

12  50 

4  io  .4 

19  40 

2  41  .6 

27  40 

i  So  -7 

43  20 

1  -7 

82  o 

8  .2 

4  ° 

11  45 

8  55 

555  -7 

12  55 

4  8  .8 

19  So 

2  40  .2 

27  50 

I  50  .0 

43  40 

I  .0 

83  o 

7  -2 

5  o 

952 

9  ° 

552  .6 

13  o 

4  7  .2 

20   0 

238  .8 

28  o 

149  .2 

44  o 

o  -3 

84  o 

6  .1 

5  5 

9  44  .0 

9  5 

549  -6 

13  5 

4  5  -6 

20  10 

2  37  -4 

28  20 

i  47  -7 

44  20 

o  59  .6 

85  o 

o  5  .1 

15  I0 

936  .2 

9  10 

546  .6 

13  io 

4  4  .1 

20  20 

2  36  .0 

28  40 

I  46  .2 

44  4° 

58  -9 

86  o 

4  -1 

J5  T5 

9  28  .6 

9  15 

543  -6 

13  is 

4  2  .6 

2O  30 

234  -6 

29  o 

i  44  .8 

45  o 

58  .2 

87  o 

3  -i 

5  20 

9  21  .2 

9  20  5  40  .7 

13  20 

4  i  .0 

20  40 

2  33  -3 

29  2O 

143  -4 

45  20 

57  -6 

88  o 

2  .O 

5  25 

9  14  .0 

9  25  iS  37  -9 

13  25 

359  -6 

20  50 

2  32  .0 

29  40 

i  42  .0 

45  4o 

56  -9 

89  o 

I  .0 

5  30 

9  7  .0 

9  30 

535  -i 

13  3° 

3  58  .1 

21   O 

2  30  -7 

30  oo 

i  40  .6 

46  o 

56  .2 

90  o 

o  .0 

!5  35 

9  o  .1 

9  35 

532  .4 

»3  35 

356  .6 

21  IO 

2  29  .4 

30  20 

139  -3 

46  20 

o  55  -6 

5  40 

853  -4 

9  40  '5  29  .6 

13  4° 

3  55  -2 

21  20 

2  28  .4 

30  40 

138  .0 

46  40 

55  -o 

5  45 

846  .8 

9  45 

527  -o 

13  45 

353  -7 

21  30 

2  26  .9 

31  o 

i  36  .7 

47  o 

54  -3 

5  5^ 

840  .4 

9  So 

524  -3 

13  5° 

21  40 

225  .7 

31  2O 

i  35  -5 

47  20 

53  -7 

5  55 
6  o 

834  -2 

8  28  .1 

9  55 

IO   O 

5  21  .7 
5  19  -2 

13  55 
14  o 

350  -9 
349  -5 

21  50 
22   0 

2  24  .5 
223  -3 

31  4° 
32  o 

i  34  -2 

133  -0 

47  40 
48  o 

53  -i 

52  -5 

• 

6  5 

8  22  .1 

10  5 

5  16  .7 

14  io 

346  .8 

22  10 

2  22  .1 

32  20 

i  31  .8 

49  ° 

050  .6 

6  10 

8  16  .2 

10  10 

5  14  -2 

14  20 

3  44  -2 

22  20 

2  20  .9 

32  4° 

1  3°  -7 

50  o 

48  .9 

6  15 

8  10  .4 

10  15  5  ii  .7 

M  3° 

3  41  -6 

22  30 

19  .8 

33  o 

i  29  .5  51  o 

47  -2 

6  20 

8  4  -8 

10  20  |5  9  .3 

14  40 

3  39  -o 

22  40 

18  .7  33  20 

i  28  .4  52  o 

45  -5 

6  25 

759  -3 

10  25 

5  6  .9 

14  50 

336  -5 

22  50 

17  -5 

33  40 

i  27  .3  53  o   43  .9 

6  30 

753  -9 

10  30 

5  4  -6 

*5  ° 

3  34  -i 

23  o 

16  .4 

34  o 

i  26  .2  54  o 

42  -3 

635 

748  -7 

io  35 

5  2  .3 

15  io 

3  31  -7 

23  io 

15  -4 

34  20 

i  25  .1  55  o  o  40  .8 

6  40 

7  43  -5 

10  40  j  5  o  .0 

15  20 

329  -4 

23  2O 

H  -3 

34  4° 

i  24  .1  56  o  j  39  .3 

6  45 

738  -4 

io  45  J4  57  -8 

15  3° 

327  -I 

2*  3° 

13  -31 

35  o 

i  23  .1  57  o   37  .8 

6  50 

733  -5 

io  50 

455  -6 

15  40 

324  .8 

23  4° 

12  .2  35  20 

I  22  .O  58   O 

36  -4 

6  55 

7  28  .6 

io  55 

453  -4 

15  5° 

3  22  .6 

23  50 

II  .2 

35  4° 

i  21  .0  59  o  i  35  .0 

7  o 

723  -8 

II   0 

16  o 

3  20  .5 

24  o 

10  .2 

36  o 

I  20  .1 

60  o 

33  -6 

7  5 

7  19  .2 

ii  5 

449  •' 

16  io 

318  .4 

24  io 

9  .2 

36  20 

1  19  .1 

61  o 

032  -3 

7  10 

714  .6 

II  10 

447  -o 

16  20 

3  16  .3 

24  20 

8  .2 

36  40 

I  l8  .2 

62  o 

31  -o 

7  i5 

7  10  .1 

"  15 

4  44  -9 

16  30 

3  14  -2 

24  3° 

7  .2 

37  o 

I  17  .2 

63  o 

29  -7 

7  20 

7  5  -7 

II  20 

442  -9 

16  40 

3  12  .2 

24  40 

6  .2 

37  20 

xx6  .3 

64  o 

28  .4 

7  25 

7  i  -4 

II  25 

4  4°  -9 

16  50 

3  io  .3 

24  50 

5  -3 

37  40 

i  15  -4 

65  o 

27  .2 

7  30 

657  -i 

II  30 

438  -9 

17  o 

3  8  .3 

25  o 

4  -4 

38  o 

I  14   -5 

66  o 

25  -9 

7  35 

653  -o 

"  35 

4  36  -9 

17  io 

3  6  .4 

25  io 

3  -4 

38  20 

i  13  -6 

67  o 

024  .7 

7  40 

648  .9 

ii  40 

435  -0 

17  20 

3  4  -6 

«5  20 

2  -5 

38  40 

I  12  .7 

68  o 

23  .6 

7  45 

644  .9 

11  45 

4  33  -i 

17  3° 

3  2  .8 

25  3° 

i  .6 

39  o 

I  II  .9 

69  o 

22  .4 

7  5° 

6  41  .0 

ii  50 

43'  -2 

17  40 

3  i  .0 

25  4° 

o  .7 

39  20 

I  II  .0 

70  o 

21  .2 

7  55 
8  o 

637  -i 
6  33  -3 

11  55 

12   O 

429  .4 
4  27  -5 

17  50 
18  o 

259  -2 

257  -5 

25  50 
26  o 

59  .8 
58  -9 

39  40 
40  o 

I  10  .2 

i  9  .4 

71  o 
72  o 

20  .1 

18  .9 

8  5 

6  29  .6 

'2   5 

4  25  .7 

18  io 

255  -8 

26  io 

58  .i 

40  20 

i  8  .6 

73  ° 

o  17  .8 

8  10 

6  25  .9 

i  12  IO 

4  23  .9 

18  20 

2  54  -i 

26  20 

57  -2 

40  40 

i  7  -8 

74  o 

16  .7 

8  15 

6  22  .3 

12  IS 

422  .2 

18  30 

252  .4 

26  30 

56  .4 

41  o 

i  7  .0 

15  .6 

8  20 

6  18  .8 

12  20 

4  20  .4 

18  40 

2  50  .8 

26  40 

55  -5 

41  2O 

I   6  .2 

76  o 

i4  -5 

825 

615  -3 

12  25 

418  .7 

18  50 

2  49  .2 

26  50 

54  -7 

41  40 

i  5  -4 

77  o 

13  5 

8  30 

6  ii  .9 

12  30 

4  17  -o 

19  o 

247  -7 

27  o 

53  -9 

42  o 

i  4  .7 

78  o 

12  .4 

8°35' 

6'  8"?5 

"«35< 

4'i5".3 

19°  io' 

,'<*», 

27°  io' 

.'53".- 

42°20' 

i'  3"-9 

79°  o' 

o'n".3 

TABLE  II  B. 


TABLE  //  D. 


629 


FACTOR  DEPENDING  ON 
BAROMETER. 


FACTOR  DEPENDING  ON  DETACHED  THERMOMETER. 


In- 

ches. 

B 

log  £ 

27.5 
27.6 

.917 
.920 

9.9622 
9-9638 

27.7 

•923 

9.9653 

27.8 

.927 

9.9609 

27.9 

•  930 

9.9685 

28.0 

•933 

9.9700 

28.1 

•937 

9.9716 

28.2 

.940 

9  973i 

28.3 

•943 

9-9747 

28.4 

•947 

9.9762 

28.5 

•95° 

9-9777 

28.6 

•953 

9-9792 

28.7 

•957 

9.9808 

28.8 

.960 

9  9823 

28.9 

•963 

9.9838 

29.0 

.967 

9-9853 

29.1 

.970 

9.9868 

2Q.2 

•973 

9  •  9883 

29-3 

•977 

9  9897 

29.4 

.980 

9  9912 

29-5 

.983 

9-9927 

29.6 

.987 

9.9942 

29.7 

.990 

99956 

29.8 

•993 

9.9971 

29.9 

997 

9.9986 

30.0 

.000 

.0000 

3O.I 

.003 

.0014 

30.2 

.007 

.0029 

30-3 

.010 

.0043 

30-4 

.013 

.0057 

30-5 

.017 

.0072 

3O.6 

.020 

.0086 

30-7 

.023 

.0100 

30  8 

027 

.0114 

3°  9 

030 

.0128 

31-0 

•033 

.0142 

TABLE  II  C. 

FACTOR  DEPENDING 

ON  ATTACHED 

THKRMOMETER. 

F 

t 

log* 

-  30° 

.007 

.0031 

—  20 

.006 

.0027 

—  IO 

.005 

.0023 

0 

.005 

.0020 

fio 

•f  20 

.004 
.003 

.0016 
.0012 

3° 

.002 

.0009 

40 

.001 

.0005 

50 

.000 

.0000 

60 

.999 

9.9996 

70 

.998 

9.9992 

80 
90 

100 

•997 
.996 
.996 

9.9989 
9.9985 
9.9981 

F 

Ll 

log  T\ 

F 

T 

log  T 

F 

r 

log  7- 

-  25° 

172 

.0688 

15° 

•°73 

.0308 

5S° 

.990 

9.9958 

24 

.169 

.0678  i 

16 

.071 

.0298 

56 

.988 

9-9949 

23 

.166 

.0669 

17 

.069 

.0289 

57 

986 

9.9941 

22 

.164 

.0658 

18 

.067 

.0280 

58 

985 

9-9933 

•  21 

.161 

.0648 

J9 

064 

.0271 

59 

983 

9.9924 

2O 

.158 

-0639 

20 

.062 

.0262 

60 

98i 

9.9916 

19 

.156 

.0629 

21 

.060 

-0253 

61 

•979 

9.9908 

18 

•  153 

.0619 

22 

058 

.0244 

62 

977 

9  9899 

17 

•  151 

.0609 

23 

.056 

•0235 

63 

•975 

16 

.148 

•  0590 

24 

°54 

0226 

64 

•973 

9.9883 

15 

•  MS 

.0590  ; 

25 

.051 

.0217 

65 

.972 

9-9875 

M 

-M3 

.0580 

26 

.049 

.0209 

66 

•97° 

9  9866 

J3 

.140 

.0570 

27 

.047 

.0200 

67 

.968 

9-9858 

12 

•  138 

.0561 

28 

•045 

.0191 

68 

.966 

9.9850 

II 

•'35 

•0551 

29 

•043 

.0182 

69 

.964 

9.9842 

IO 

•133 

.0541 

3° 

.041 

.0173 

70 

.962 

9-9834 

9 

.130 

.0532 

31 

•°39 

.0164 

71 

.961 

9.9825 

8 

.128 

-0522 

32 

•036 

•0155 

72 

•959 

9.9817 

7 

•125 

•0513 

33 

•034 

.0147 

73 

•957 

9  9809 

6 

.123 

•0503 

34 

.032 

.0138 

74 

•955 

9.9801 

5 

.120 

.0494 

35 

.030 

.0129 

75 

•953 

9  9793 

4 

.Il8 

.0484 

36 

.028 

.0120 

76 

•  952 

9-9785 

3 

•TI5 

0475 

37 

.026 

.OII2 

77 

-95° 

9-9777 

2 

•"3 

.0465 

38 

.024 

.0103 

78 

.948 

9  9769 

—   I 

.in 

.0456 

39 

.022 

.0094 

79 

.946 

9  976i 

0 

.108 

.0446 

40 

.O2O 

.0086 

80 

•945 

9  9753 

+   I 

.106 

•°437 

41 

.Ol8 

.0077 

81 

•943 

9-9745 

2 

.103 

.0428 

42 

.016 

.0068 

82 

.941 

9-9737 

3 

.  IOI 

.0418 

43 

.014 

.0060 

83 

•939 

9.9729 

4 

099 

.0409 

44 

.012 

.00^1 

84 

•938 

9.9721 

5 

.096 

.0400 

45 

.010 

.0043 

85 

•936 

9-97J3 

6 

.094 

.0390 

46 

.008 

.0034 

86 

•934 

9-9705 

7 

.092 

.0381 

47 

.006 

.0026 

87 

•933 

9.9697 

8 

.089 

.0372 

48 

.004 

.0017 

88 

•931 

9.9689 

9 

.087 

•0363 

49 

.002 

.0009 

89 

.929 

9.9681 

IO 

.085 

•0353  i 

50 

.000 

.0000 

90 

.928 

9.9673 

ii 

.082 

•0344 

Si 

.998 

9.9992 

9T 

.926 

9-9665 

12 

.080 

•°335 

52 

.996 

9.9983 

92 

.924 

9.9658 

13 

.078 

.0326 

53 

•994 

9  9975 

93 

•923 

9.9650 

14 

.076 

•0317 

54 

•992 

9.9966 

94 

.921 

9.9642 

+  is 

•°73 

.0308 

55 

.990 

9.9958 

95 

.919 

9.9634 

r  =  (mean  refraction)  XSX 


630 


TABLE  III  A. 

BESSEL'S  REFRACTION  TABLE. 


ll 

8s  i 

c  <u 

Si's 

If  i 

II 

rt  'c  rt 

log  a. 

Dif. 

A. 

A. 

«  3 

cS'a  § 
&S.2 

log  a. 

Dif 

A. 

A. 

•8*°  5 

<< 

<  Q 

5°  o 

85°  o 

i  .  71020 

i  .0127 

1.1229 

14°  20 

75°  40 

I-7539I 

.0212 

10 

84  50 

1.71279 

259 

243 

I.OI2I 

1.1178 

3° 

3° 

i  .  75408 

17 

-0208 

20 

3° 

1.71522 

^43 
227 

I  .0115 

1.1130 

40 

20 

1-75425 

16 

.O2O4 

30 

40 

1.71749 

212 

I  .OIIO 

1.1082 

50 

10 

!•  75441 

16 

.0200 

40 

20 

1.71961 

1.0105 

1.1036 

15    0 

75   o 

i-  75457 

Q/r 

.0197 

.  50 
6   o 

IO 

84   o 

i  .  72160 
1.72346 

199 

186 

1  .0100 
1.0096 

1.0992 
1-0951 

16 

74 
73 

1-75543 

OO 
72 

•0175 
.0156 

10 

20 

83  50 

40 

1.72519 
1.72681 

i73 
162 

I  .0092 

1.0088 

1.0914 
i  0879 

18 
19 

72 

i  75675 
1.75726 

51 

0139 
.0124 

3° 

3° 

1.72832 

151 

1.0084 

1.0846 

20 

70 

I-7577I 

45 

.OIII 

40 

20 

1.72974 

142 

1.0081 

1.0815 

21 

69 

i  .  75809 

'  38 

.OIOI 

50 

IO 

1-73105 

131 

1.0078 

1.0784 

22 

68 

1.75842 

33 

.0092 

7   o 

83  o 

i  .  73229 

124 
1  18 

1.0075 

1-0754 

23 

67 

1-75871 

26 

.0083 

IO 

82  50 

1-73347 

112 

1.0073 

1.0725 

24 

66 

I-75897 

20 

.0075 

20 

40 

1-73459 

I  .0070 

1.0697 

25 

65 

1.75919 

22 

.0068 

30 

30 

1  73564 

1°5 

I  0067 

i  .0671 

26 

64 

1-75939 

TQ 

.0063 

40 

20 

1-73663 

99 

1.0065 

i  .0646 

27 

63 

1-75957 

IO 

16 

.0058 

o  5° 
8   o 

IO 

10 

82   o 
81  50 

1-73757 
1-73845 
1.73928 

94 
88 
83 

I  .0062 
I  .0060 

1.0058 

1.0622 
i  .  0600 
1.0579 

28 
29 
30 

62 
61 
60 

1-75973 
1.75988 
i  .  76001 

15 

.0054 
0049 
0046 

20 

40 

i  .  74007 

79 

I  .0056 

31 

59 

i  .  76012 

11 

0043 

3° 

30 

1-74083 

70 

1.0054 

1.0540 

32 

58 

1.76023 

ii 

0040 

40 

20 

1-74I55 

68 

1.0052 

1.0523 

33 

57 

1-76033 

IO 

0037 

50 

10 

1.74223 

1.0050 

1.0508 

34 

56 

i  .  76042 

9 

0034 

9   ° 

81   o 

1.74288 

g5 

1.0049 

1-0493 

35 

55 

i  .  76050 

g 

0031 

i    I0 

80  50 

I-74352 

60 

1.0047 

1.0479 

36 

54 

1.76058 

0029 

20 

30 

40 
30 

1.74412 
1.74468 

56 

1.0046 
1.0045 

i  .  0466 

ll 

53 

52 

1.76065 
1.76071 

7 

6 

0027 
0026 

40 

20 

1.74521 

53 

1.0043 

1.0442 

39 

51 

r.  76077 

0025 

50 

10 

1-74573 

52 

I  .0042 

1.0431 

40 

5° 

1.76082 

5 

0023 

10   0 

So  o 

1.74623 

5° 

I  .0041 

i  .  0420 

41 

49 

1.76087 

5 

OO2I 

IO 

79  So 

1.74670 

47 

1.0040 

1.0409 

42 

48 

1.76092 

5 

OO2O 

20 

40 

1.74714 

44 

I  0039 

1.0398 

43 

47 

i  .  76096 

4 

0019 

30 

3° 

1-74757 

43 

1.0038 

1.0387 

44 

46 

1.76100 

4 

OOIQ 

40 

20 

i  74799 

42 

1.0037 

45 

45 

1.76104 

4 

00l8 

50 

10 

1.74839 

4° 

I  .0036 

1.0367 

46 

44 

1.76107 

3 

II    0 

79   o 

1.74876 

37 
16 

1-0035 

1-0357 

47 

43 

1.76111 

4 

10 

78  50 

1.74912 

3° 

i.ooj4 

J-0347 

48 

42 

1.76114 

3 

20 

40 

i  .  74947 

35 

1-0033 

1-0338 

49 

1.76117 

3 

30 

30 

1.74981 

34 

i  .0032 

1.0328 

50 

40 

1.76119 

2 

40 

20 

i-75OI3 

32 

i  .0031 

i  .  03  i  8 

51 

39 

1.76122 

3 

5° 

IO 

i  •  75043 

3° 

1.0030 

i  .  0308 

52 

38 

1.76124 

2 

12   0 

78   o 

1.75072 

29 

i  .0030 

i  0299 

53 

37 

i  .76126 

2 

10 

77  So 

1.75101 

29 

1.0029 

i  .  0290 

54 

36 

1.76128 

2 

20 

4° 

1.75129 

26 

i  .0028 

i  0281 

65 

35 

1.76130 

2 

30 

30 

i  .0027 

i  .0272 

56 

34 

1.76132 

2 

40 

20 

i  .75180 

25 

1.0027 

i  0264 

57 

33 

1.76134 

2 

50 

10 

i  •  75205 

25 

1.0026 

1.0258 

58 

32 

1.76136 

2 

13   o 

77   o 

1.75229 

24 

i  .0026 

i  .0252 

59 

31 

1.76138 

2 

10 

76  50 

1-75252 

23 

1.0246 

60 

3° 

1.76139 

* 

20 

40 

J-  75274 

22 

i  .0241 

65 

25 

1.76145 

3° 

30 

1-75295 

21 

1.0235 

70 

20 

1.76149 

4 

4° 

20 

I-753I6 

21 

i  .  0230 

75 

15 

1-76152 

3 

5° 

10 

1.75336 

2O 

1.0225 

80 

IO 

1.76154 

2 

14  o 

76   o 

1-75355 

Is 

i  .0220 

85 

5 

1.76156 

2 

10 

75  50 

1-75373 

18 

i  .0216 

90° 

o»  o' 

1.76156 

I4»  20' 

75°  40' 

i  .0212 

TABLE  III.  A. 


TABLE  III  D. 


63 1 


SUPPLEMENT. 


FACTOR  DEPENDING  ON  DETACHED 
THERMOMETER. 


Apparent 
Altitude. 

Apparent 
Zenith 
Distance. 

Logarithm 
of 
Refraction. 

A. 

A. 

o°   30' 

89°  30' 

3-24142 

.0780 

.5789 

I           0 

89     t> 

3.16572 

•°593 

4653 

i     30 

88   30 

3.09723 

.0465 

•3797 

2          0 

88     o 

3.03686 

.0368 

•SM1 

2        30 

87   30 

2.98269 

.0298 

.2624 

3       o 

87     o 

2.93174 

.0244 

.2215 

3     3° 

86  30 

288555 

.0204 

.1888 

4       ° 

86     o 

2.84444 

.0172 

.1624 

4     3° 

85   30 

2.80590 

.0147 

.1408 

5       o 

85     o 

2.76687 

.0127 

.1229 

TABLE  III  B. 


FACTOR  DEPENDING 
ON  BAUOMETEK. 

Inches. 

Log  B. 

27-5 

—  .03191 

27.6 

-•03033 

27.7 

—  .02876 

27.8 

—  .02720 

27.9 

—  .02564 

28.0 

—  .02409 

28.1 

—  .02254 

28.2 

-.02099 

28.3 

—  .01946 

28.4 

—  .01793 

28.5 

—  .01640 

28.6 

—  .01488 

28.7 

—  01336 

28.8 

—  .01185 

28.9 

-  01035 

29.0 

—  .00885 

29.1 

—  .00735 

29.2 

-  .00586 

29-3 

—  .00438 

29.4 

—  00290 

29-5 

—  .00142 

29.6 

+  .00005 

29.7 

.00151 

29.8 

.00297 

29.9 

.00443 

30.0 

.00588 

30.1 

.00732 

30.2 

.00876 

3°  3 

.01020 

30.4 

.01163 

30.5 

.01306 

30.6 

.01448 

30-7 

.01589 

30.8 

.01731 

3°-9 

.01871 

31-0 

.02012 

TABLE  III  C. 

FACTOR  DEPENDING  ON 
ATTACHED  THERMOMETER. 


F. 

Log/. 

-  30° 

-.00242 

—  20 

-.00203 

—  10 

-.00164 

O 

-.00125 

+  10 

-.00086 

20 

- 

-.00047 

3° 

+  .00008 

40 

-.00031 

50 

-  .00070 

60 

-.00100 

70 

-  .00148 

80 

-.00186 

90 

-  .00225 

loo 

-  .00264 

log  |8  =  log  B  +  log  *. 


F. 

Logy. 

F. 

Logy. 

F. 

Logy. 

-25° 

+  .06773 

15 

+  .02969 

55 

—  .00528 

-24 

.06674! 

16 

.02878 

56 

—  .00612 

—23 

•06575 

17 

.02787 

57 

-.00698 

—  22 

.06476; 

18 

.02697 

58 

—  .00780 

—  21 
—  2O 

.  06377  j 
.06279 

20 

.02606 
.02514 

59 
60 

-.00863 
—  .00946 

-19 

.06181 

I 

.02426 

61 

—  .01029 

-18 

•06083 

2 

.02336 

62 

—  .on  12 

-17 

•05985 

3 

.02247 

63 

-.01195 

-16 

.05887 

4 

•02157 

64 

—  .01278 

-15 

.05790 

.02068 

—  .01360 

•05693 

6 

.01979 

66 

—  .01443 

—  13 

•05596 

7 

.0*890: 

67 

-.01525 

—  12 

.05500 

8 

.01801  J 

68 

—  .0:607 

—  II 

.05403 

29 

•01713! 

69 

—  .01689 

—  10 

•05307 

3° 

.01624' 

70 

—  .01770 

—  9 

05211 

3' 

.01536 

-.01852 

-  8 

.05115 

32 

.01448 

72 

—  •01933 

—  7 

.05020 

33 

.01360 

73 

—  .02015 

—  6 

.04924 

34 

.01273 

74 

—  02096 

-  5 

.04829 

35 

.01185 

75 

—  .02177 

-  4 

•04734; 

36 

.01098 

76 

—  .02257 

—  3 

.04640 

37 

.OIOII 

77 

-.02338 

—  2 

•04545 

38 

.00924 

78 

—  .02419 

—  i 

.04451 

39 

.00837 

79 

—  .02499 

—  0 

•04357' 

40 

.00750 

80 

-.02579 

+  i 

.04263 

41 

.00664 

81 

—  .02659 

+  2 

.04169 

42 

.00578 

82 

—  .02738 

-j—   5 

.04076 

43 

.00492, 

83 

—  .02819 

+  4 

.03982 

44 

.  00406 

84 

—  .02898 

+  5 
+  6 

.03889 
.03796 

a 

.00320 
.00234 

88i 

—  .02978 
—  .03057 

+  7 

.03704 

47 

00149: 

87 

—  .03136 

+  8 

•03611; 

48 

+.00064! 

88 

—  .03216 

+  9 

.035  IQ 

49 

—  .00021 

89 

—  .03294 

+  10 

•03427; 

So 

—  .OOIO6 

90 

--03373 

+11 

•03335 

51 

—  .OOlgi 

-•03452 

+  12 

•03243 

52 

—  .00275 

92 

—  .0353° 

+13 

•03152 

53 

—  .00360 

93 

-.03609 

+14 

.03060 

54 

—  .00444 

94 

—  .03687 

+15 

+.02969 

55 

—  .00528 

95 

—  .03765 

log  r  =  log  a  +  A  .  log  /3  +  A  .  log  y  +  log  tan  *. 


632 


TABLE  IV. 


To  CONVERT  CENTIMETRES  INTO  INCHES. 


il 

II 

•  w 

£  v 
C  i; 

qj  t"? 

"Sg 

•    ai 

•£  <u 
c  i 

|| 

us 

§  = 

us 

«~ 

38 

II 

68.0 

26.772 

73-5 

28.938 

O.I 

•0394 

68.5 

26.969 

74.0 

29-J34 

O.2 

.0787 

69.0 

27.166 

74-5 

29-331 

0  3 

.1181 

69-5 

27-363 

29.528 

0.4 

•I575 

70.0 
70.5 

27.560 
27.756 

75-5 
76.0 

29-725 
29.922 

°'5 
0.6 

.1969 
.  2362 

71.0 

27-953 

76.5 

30.119 

o  7 

.2756 

7i.5 

28.150 

77.0 

3°-3'6 

0.8 

•3*5° 

72.0 

28.347 

77-5 

30.512 

0.9 

•3543 

72-5 
73-o 

28.544 
28.741 

78  o 
78.5 

30  709 
30.906 

1.0 

•3937 

TABLE  V. 

To  CONVERT  READING  OF  CENTIGRADE 
THERMOMETER  INTO  FAHRENHEIT'S. 


TABLE  VI. 

To  CONVERT  RKADING  OF  RKACMUR'S 

THEKMOMEThK   INTO    FAHRKNHHT's. 


C. 

F. 

C.  , 

F. 

C. 

F. 

—32° 

-25°.  6 

+3° 

+37°  -4 

o.t 

~8 

—  31 

-23  8 

4 

39  -2 

.2 

•36 

^O 

—  22  .O 

5 

41  .0 

-3 

•54 

—29 

—  20  .2 

6 

42  .8 

•4 

•7' 

T28 

—27 

-18  .4 
-16  .6 

7 
8 

44  .6 
46  -4 

•5 
.6 

.90 
.08 

—26 

-14  .8 

9 

48  .2 

•7 

.26 

—25 

—  13  .0 

10 

50  .0 

.8 

•44 

-24 

—  II  .2 

ii 

51  .8 

•9 

.62 

-23 

-9-4 

12 

53  6 

i  .0 

.80 

—  22 

-  7  -6 

13 

55  -4 

—  21 

-  5  -8 

57  -2 

—  2O 

—  4  .0 

15 

59  -° 

—  19 

—  2  .2 

16 

60  .8 

-18 

—  o  .4 

*7 

62  .6 

—  17 

18 

64  .4 

-16 

4-  3  -2 

19 

66  .2 

—15 
—  14 

-fs  -o 
-|-  6  .8 

20 
21 

68  .0 
69  .8 

-i-  8  .6 

22 

71  .6 

—  12 

10  .4 

23 

73  -4 

—  II 

12  .2 

24 

75  -2 

—  10 

--1 

H  -o 
IS  -81 
17  .6 

11 

27 

77  -o 
78  .8 
80  .6 

—  7 

28 

82  .4 

-  6 
—  5 

21  .2 

23  -o 

29 
30 

84  .2 

86  .0 

—  4 

24  .8 

31 

87  -8 

—  3 

26  .6 

32 

89  .6 

28  .4 

33 

9i  -4 

—  i 

3°  2 

34 

93  -2 

0 

32  .0 

35 

95  -^ 

4-  i 

33  -8 

36 

96  .8 

2 

35  -6 

37 

98  .6 

3 

37  -4 

38 

100  .4 

R. 

F. 

R. 

F. 

R. 

F. 

—25° 

—24°.  25 

3° 

38°.  75 

0°.I 

°.225 

—24 

—  22  .O 

4 

41  .0 

O  .2 

•45 

-23 

—19  -75 

5 

43  25 

o  -3 

•675 

-22 

—  17  -5 

6 

45  -5 

o  .4 

.90 

—  21 

—  15  -251 

7 

47  -75 

o  -5 

•'25 

—  20 

—13  .0  ! 

8 

50  .0 

o  .6 

•35 

—  19 

—  10  .75 

9 

52  .25 

o  .7 

•575 

-18 

-  8  .5 

10 

54  -5 

o  .8 

.80 

-17 

-  6  .25 

ii 

56  -75 

o  .9 

.025 

-16 

—  4  .0 

12 

59  ° 

I  .0 

•25 

—15 

-  i  -75 

13 

61  .25 

—  M 

f  o  5 

H 

63  -5 

-'3 

2  -75 

65  -75 

—  12 

5  ° 

16 

68  .0 

—  II 

7  -25 

»7 

7°  -25 

—  10 

9  -5 

18 

72  -s" 

—  9 

ii  .75 

!9 

74  -75 

—  8 

14  .0  ; 

20 

77  .0 

—  7 

16  .25 

21 

79  -25 

-  6 

18  .5 

22 

81  5 

-  5 

20  .75 

23 

83  -75 

—  4 

23  .0 

24 

86  .0 

-  3 

25  -25 

25 

88  .25 

—  2 

27  -5  i 

26 

9°  -5 

—  1 

29  -75 

27 

92  -75 

—  0 

32  .0 

28 

95  -° 

4-  i 

34  25 

29 

97  25 

2 

36  -5 

3° 

99  -5 

3 

38  -75 

3i 

101  .75 

TABLE    VII. 


633 


To  CONVERT  HOURS,  MINUTES,  AND  SECONDS  INTO  A  DECIMAL  OF  A  DAY. 


Hour. 

Decimal  of 
Day. 

Minute. 

Decimal  of 
Day. 

Second. 

Decimal  of 
Day. 

i 

.041  6667 

i 

.000  6944 

i 

.000  0116 

2 

•083  3333 

2 

.001  3889 

2 

.000  0231 

3 

.125  oooo 

3 

.002  0833 

3 

.000  0347' 

4 

.166  6667 

4 

.002  7778 

4 

.000  0463 

I 

•208  3333 
.250  oooo 

| 

.003  4722 
.004  1667 

5 
6 

.000  0579 
.000  0694 

7 

.291  6667 

7 

.004  8611 

7 

.000  0810 

8 

•333  3333 

8 

•005  5556 

8 

.000  0926 

9 

•375  o000 

9 

.006  2500 

9 

.000  1042 

10 

.416  6667 

10 

.006  9444 

10 

.000  1157 

ii 

•458  3333 

ii 

.007  6389 

ii 

.000  127^ 

12 

.500  oooo 

12 

•008  3333 

12 

.000  1389 

J3 
J4 

.541  6667 

•583  3333 

J3 
*4 

.009  0278 
.009  7222 

13 
'4 

.000  1505 
.000  1620 

15 

.625  oooo 

15 

.010  4167 

15 

.000  1736 

16 

.666  6667 

16 

.Oil  IIII 

16 

.000  1852 

J7 

•7°8  3333 

17 

.on  8056 

T7' 

.000  1968 

18 

.750  oooo 

18 

.012  5000 

18 

.000  2083 

!Q 

.791  6667 

19 

.013  1944 

J9 

.OOO  2lOg 

20 

•833  3333 

20 

.013  8889 

.20 

.000  2315 

21 

.875  oooo 

21 

.014  5833 

21 

.000  2431 

22 

.916  6667 

22 

.015  2778 

22 

.000  2546 

23 

.958  3333 

23 

•015  9722 

23 

.000  2662 

24 

I  .OOO  OOOO 

24 

.0  6  6667 

24 

.000  2778 

25 

.0  7  3611 

2| 

.000  2894 

26 

.0  8  0556 

26 

.000  3009 

27 

.0  8  7500 

27 

.000  3121; 

28 

.0  9  4444 

28 

.000  3241 

29 

.020  1389 

29 

.000  3356 

3° 

.020  8333 

3° 

.000  3472 

31 

.021  5278 

3i 

.000  3588 

32 

.022  2222 

S2 

.000  3704 

33 

.022  9167 

33 

.000  3819 

34 

.023  6111 

34 

•ooo  3935 

P 

.024  3056 

.025  oooo 

35 
36 

.000  4051 
.000  4167 

37 

.025  6944 

37 

.000  4282 

38 

.026  3889 

38 

.000  4398 

39 

.027  0833 

39 

.000  4514 

40 

.027  7778 

40 

.000  4630 

4i 

.028  4722 

41 

.000  4745 

42 

.029  1667 

42 

.000  4861 

43 

.029  8611 

43 

.000  4977 

44 

•°3°  5556 

44 

.000  5093 

45 

031  2500 

45 

.000  5208 

46 

.031  9444 

46 

.000  5324 

47 

.032  6389 

47 

.000  5440 

48 
49 

•°33  3333 
.034  0278 

48 
49 

.000  5556 
.000  5671 

50 

.034  7222 

5° 

.000  5787 

Si 

.035  4167 

5i 

.000  5903 

52 

.036  mi 

52 

.000  6019 

53 

.036  8056 

53 

.000  6134 

54 

.037  5000 

54 

.000  6250 

55 

.038  1944 

55 

.000  6366 

56 

.038  8889 

56 

.000  6481 

57 

•°39  5833 

57 

.000  6507 

58 

.040  2778 

58 

.000  6713 

59 

.040  9722 

59 

.000  6829 

60 

.041  6667 

60 

.000  6944 

L_ 

634 


TABLE    VIII  A. 


sin  i 


iy 

0 

m 

I 

m 

2 

m 

3 

m 

M 

log  m 

m 

logm 

•m 

log  m 

Ml 

log  m 

0 

i 

2   • 

3 
4 

o".oo 

0  .00 
0  .00 

o  .00 

0  .01 

6-73673 
7-33879 
7.69097 
7.94085 

".96 
•°3 
.10 
.16 
•  23 

.29303 
•30739 
•32151 
•33541 
.34909 

7"-85 
7  -98 

8  .12 

8  .25 
8  -39 

.89509 
.90230 
.90945 
.91654 
•92357 

17".  67 

17  -87 
18  .07 
18  27 
18  .47 

1.24727 

1.25208 

1.25687 

1.26163 

1.26636 

I 

9 

o  .01 

0  .02 

o  .02 
o  .03 
o  .04 

8.13467 
8.29303 
8.42692 
8.54291 
8  64521 

:U 

•45 
•52 
.60 

•36255 
•3758i 
.38888 
.40174 
.41442 

8  .52 
8  .66 
8  .80 
8  -94 
9  .08 

•93055 
•93747 
•94434 
•95H5 
•95791 

18  .67 
18  .87 
19  .07 
19  .28 
19  .48 

1.27107 

1-27575 
1.28041 

1.28504 

1.28965 

10 

ii 

12 
13 
14 

o  .05 
o  .06 
o  .08 
o  .09 

0  .11 

8-73673 
8.81951 
8.89509 
8.96461 
9.02898 

.67 

% 

.91 
•99 

.42692 
•43925 
.45140 
•46338 
•47519 

9  .22 

9  -36 
9  -50 
9  -64 
9  -79 

.96462 
.97127 
.97788 

•98443 
.99094 

19  .69 
19  .90 

20  .11 
2O  .32 
20  .53 

1.29423 
1.29879 
1-30332 
1.30783 
1.31232 

16 
17 
18 
19 

0  .12 

o  .14 
o  .16 
o  .18 
o  .20 

9  08891 
9.14497 
9.19763 
9  24727 
9.29423 

3  -°7 
3  -IS 
3  -23 
3  -32 
3  -40 

.48685 
.49836 
•50971 
.52092 
•53r98 

9  -94 
10  .09 
10  .24 

10  .39 

10  .54 

•99740 
1.00381 
1.01017 
1.01649 
1.02276 

20  .74 
20  .95 
21  .16 
21  .38 
21  .60 

1.31679 
1.32123 
1.32566 

1.33006 

t-33443 

20 

21 
22 

23 
24 

O  .22 

o  .24 
o  .26 
o  .28 
o  .31 

9-33879 
938117 
9.42157 
9.46018 
9-497I5 

3  -49 

1$ 

It 

•54291 
•55370 
.56436 
•57489 
•58529 

10  .09 
10  .84 
II  .00 

II  .15 

II  31 

1.02898 
1-03517 
1.04131 
1.04740 
i  05345 

21  .82 
22  .03 
22  .25 
22   .47 
22   .70 

133878 
I-343H 
1-34743 
i-35*72 
I-35598 

25 
26 
27 
28 
29 

o  -34 
o  -37 
o  .40 

0  -43 

o  .46 

9  5326i 
9-56067 
9-599-15 
9  63104 

9.66l  S2 

3  -94 
-03 

<   .12 
.   .22 

^   .  ?2 

•59557 
•60573 
•6t577 
•62570 
6355  ' 

II  .47 

II  .63 

II  .79 

ii  -95 

12  .  II 

1.05946 
1.0654^ 
1.07136 
1.07725 
1.08310 

22  .92 
23  -14 

23  -37 
23  .60 
23  .82 

1.36022 
i  36445 
1.36866 
1.37285 
1.37702 

30 

31 
32 
33 
34 

o  .49 
o  .52 
o  .56 
o  -59 
o  .63 

9.690^7 
9  71945 
9  74703 
9-77376 
9-79968 

.42 

•52 
.62 

| 

•64521 
.6s48i 
•66431 

•6737° 
.68299 

12  .27 
12  .43 

12  .DO 
12  ,76 

12  .93 

i  08891 
1.09468 
1.10042 
1.  10611 
1.11177 

24  .05 
24  .28 

24  -51 

24  -74 
24  .98 

138116 
1.38529 
1.38940 
I-39348 
z-39755 

13 

s 

39 

o  .67 
o  .71 
o  -75 
o  .79 
o  .83 

9.82486 

9-84933 
9.873*3 
9.89629 
9  91886 

4  .92 
5  -03 
5  -13 
5  -24 
5  -34 

.69218 
.70127 
.71027 
.719.8 
.72800 

13  .10 

13  -27 

13  -44 
13  .62 
H  -79 

1.11739 
1.12298 
1.12853 
i  13404 
I-I3952 

25  .21 

25  -45 
25  .68 
25  .92 
26  .T6 

1.40160 
1.40563 
1.40964 
1.41364 
1.41761 

40 
41 
42 
43 
44 

o  .87 
o  .91 
o  .96 

I  .01 

I  .06 

9.94085 
9  96229 
9-98323 
o  00366 
.02363 

5  -45 
5  -56 
5  67 
5  -78 
5  -90 

•73673 
•74537 
•75393 
.76240 
.  77080 

13  -96 
M  -'3 
14  -3i 
14  .49 
14  .67 

1.14497 
1.15038 
I-I5576 
1.  16110 
1.16641 

26  .40 
26  .64 
26  .88 
27  .12 
27  -37 

1-42157 
1-42551 
I-42943 
1-43333 
i  43722 

45 
46 
47 
48 
49 

T  .10 

I  -15 

I  .20 

I  .26 
1  -31 

.04315 
.06224 
.08092 
.09921 

.II7T2 

6  .01 
6  .13 
6  .24 
6  .36 
6  .48 

.77911 
•78734 
•7955° 
.80358 
.81158 

H  -85 
15  -03 

15  .21 

15  -39 
15  -57 

1.17169 
1.17694 
1.18216 
1  18735 
1.19250 

27  .61 
27  .86 
28  .10 
28  .35 
28  .60 

1.44109 

1-44494 
1.44877 

I-45259 
1-45639 

50 

Si 
52 
53 
54 

I  .36 
I  .42 

I  .48 

1  -53 

i  -59 

•  13467 

.15187 
.16875 

.18528 
.20151 

6  .60 
6  .72 
6  .84 
6  .96 
7  -09 

•81952 
.82738 
•83517 
.84788 
•85053 

15  -76 
15  -95 
16  .14 
16  .32 
16  .51 

1.19762 
1.30271 
1.20778 
1.21281 
1.21782 

28  .85 
29  .10 
29  -36 
29  .61 
29  .86 

1.46018 
1.46395 
1.46770 
I-47I43 
i-475T5 

55 
56 
57 
58 
59 

i  -65 

i  .71 
i  -77 
i  .83 
i  .89 

.21745 

.23310 
.24848 
.26358 
.27843 

7  .21 

7  -34 
7  -46 
7  .60 
7  -72 

.85813 
.86564 
•87310 
.88049 
.88782 

16  .70 
16  .89 
17  .08 
17  .28 
17  -47 

1.22280 
1.22775 
1.23267 
1.23756 
1.24243 

30  .12 
30  .38 
30  .64 
30  .90 
31  .l6 

1.47886 
1.48255 
1.48622 
1.48988 
1.49352 

60 

i".96 

.29303 

7".85  ' 

.89509 

I7".67 

i  24727 

3i"-42 

1.49714 

TABLE    VIII  A. 


635 


2  sin' ^ 


4 

m 

5 

. 

6 

m 

7 

m 

m 

log  w 

M 

log  m 

m 

log  m 

m 

log  m 

o 

I 

2 

3 
4 

3l".42 

31  .68 

3i  -94 
32  .20 

32  -47 

1.49714 

1.50076 
I-5°435 
I-50793 
1.51150 

49".09 
49  .41 
49  -74 
50  .07 
50  .40 

1.69096 
1.69185 
1.69673 
1.69960 
1.70246 

7o".68 
71  .07 
71  .47 
71  .86 
72  .26 

1.84931 

1.85172 
1.85412 
1-85651 
1.85890 

96".  20 

96  .66 
97  .12 
97  -58 
98  .04 

1.98320 
1.98526 
i  98732 
1.98937 
1.99142 

I 

7 
8 
9 

32  -74 
33  -oi 
33  -27 
33  -54 
33  -81 

i-S^oS 
i.5t859 
1.52211 
1.52562 
1.52912 

5°  -73 
51  .07 
51  .40 
5i  -74 
52  -07 

1-70531 
1.70815 
1.71099 
1.71382 
1-71663 

72  .66 
73  .06 
73  -46 
73  -86 
74  -26 

1.86129 
1.86366 
1.86603 
1.86840 
1.87075 

98  -50 
98  .97 
99  -43 
99  .90 
loo  .37 

*•  99347 
I-9955I 
J-99755 
1.99958 
2.00161 

10 
ii 

12 
13 

*4 

34  -36 
34  -64 
34  -9i 
35  -19 

1.53260 
1.53606 
1-53952 
1.54296 
1.54639 

52  .41 
52  -75 
53  -09 
53  -43 
53  -77 

1.7:944 
1.722-3 
1.72502 
1.72780 
I-73°57 

74  .66 
75  -06 
75  -47 
75  -88 
76  .29 

1.87310 

I-87545 
1.87779 
1.88012 
1.88244 

IOO  .84 

101  .31 

101   .78 
1O2  .25 
102  .72 

2.00363 
2.00565 
2.00766 
2.00967 
2.01167 

IS 

16 

J7 
18 
19 

35  -46 
35  -74 

36  .02 
36  -30 
36  .58 

1.54980 
I-5532Q 
I-55659 
1.55996 

1-56332 

54  .« 
54  -46 
54  -80 

55  -T5 
55  -50 

1  -73333 
1.73608 
1.73883 
I-74157 
i  74429 

76  .69 
77  .10 
77  5i 

77  -93 
78  -34 

1.88476 
1.88708 
1.88938 
1.89168 
1.89398 

103  .20 
103  .67 
I04  .15 
104  .63 

105  .10 

2.01367 
2.01566 
2.01765 
2.01964 
2.02162 

20 
21 

22 

23 
24 

36  .87 

37  -15 
37  -44 
37  -72 
38  01 

1.56667 
1.57000 
1-57332 
i.57663» 
1-57993 

55  -84 
56  .19 
56  -55 
56  .90 
57  -25 

1.74701 
1.74972 
1.75242 
I-755]rl 
1.75780 

78  -75 
79  -16 
79  -58 
80  .00 
.  80  .42 

1.89627 
1.89855 
1.90083 
1.90310 
1.90536 

10!  -55 

106  .06 
106  .55 
107  .03 
107  .51 

2.02360 
202557 
2.02753 
2.02950 
2.03146 

25 
26 
27 
28 
29 

38  -30 
38  -59 
38  .88 
39  -J7 
39  -46 

1-58321 
1.58648 
1.58974 
1.59299 
1.59622 

57  -60 
57  -96 
58  .32 
58  .68 
59  -03 

1.76048 
1.76314 
1.76580 
1.76846 
1.77110 

80  .84 
81  .26 
81  .68 
82  .10 
82  .52 

1.90762 
1.90987 
1.91212 
1.91436 
1.91660 

107  .99 
108  .48 
108  .97 
109  .46 
109  ,95 

2.03341 
2-03536 

2  03730 
2  03924 
2.04II8 

30 

31 
32 
33 
34 

39  -76 
40  .05 
40  .35 
40  .65 
40  -95 

T-  59945 
1.60266 
1.60586 
1.60904 
1.61222 

59  -40 
59  -75 
60  .11 
60  .47 
60  .84 

J-77373 
1.77636 
1.77898 
1.78160 
1.78420 

SMS 

83  .81 
84  .23 

84  .66 

1.91883 
1.92105 
1.92327 
1.92548 
1.92769 

no  .44 
no  .93 
in  .43 
in  .92 
112  .41 

2.043II 
2.04504 
2.04697 
2.04888 
2.05080 

11 

37 
38 
39 

4i  -25 
4i  -55 
41  .85 
42  .15 
42  -45 

1.61538 
i.6i8s4 
1.62168 
1.62481 
1.62793 

6l   .20 

61  -57 
61  .94 
62  .31 
62  .68 

1.78680 
1.78938 
1.79197 

*•  79454 
1.79710 

85  .09 
85  -52 
85  -95 
86  .39 
86  .82 

1.92990 
1.93209 
1.93428 
1.93646 
1.93864 

112  .90 
113  .40 
1I3  .00 

114  .40 
114  .90 

2.05271 
2.05462 
2.05652 
2.05842 
2.06031 

40 
41 
42 

43 
44 

42  .76 
43  -06 
43  -37 
43  -68 
43  -99 

1.63103 

I-634I3 
1.63722 
1.64029 
1-64335 

63  .05 
63  .42 
63  -79 
64  .16 

64  -54 

1.79967 
1.80221 
1.80476 
1.80729 
1.80982 

87  .26 
87  .70 
88  .14 
88  .57 
89  .01 

1.94082 
1.94299 
I.945I5 
I-94731 
1.94946 

115  .40 
115  .90 

116  .40 
116  .90 
117  .41 

2.O622O 
2.06409 
2.06597 
2.06785 
2.06972 

45 
46 
47 
48 
49 

44  -30 
44  .61 
44  -92 
45  -24 
45  -55 

1.64641 
1.64945 
1.65248 
I-655S0 
1.65851 

64  .91 

A5  'A9 
6|  -67 
66  .05 
66  .43 

1.81234 
1.81486 
1.81736 
1.81986 
1.82236 

89  -45 
89  .89 
90  -33 
90  .78 
91  .23 

i.95i6i 
1-95375 
^95589 
1.95802 
1.96014 

117  .92 
118  .43 
1.8  .94 
"9  -45 
119  .96 

2.07159 
2.07346 
2  07532  . 
2.07718 
2.07903 

50 

5i 
52 
53 
54 

45  -87 
46  .18 
46  .50 
46  82 
47  .14 

1.66151 
1.66450 
i.  66748 
1.67045 
1.67341 

66  .81 
67  .19 
67  .58 
67  .96 
68  .35 

1.82484 
1.82732 
1.82979 
1.83225 
1.83471 

91  .68 

92  .12 

92  -57 
93  .°2 
93  -47 

1.96226 
1.96438 
1.96649 
1.96860 
1.97070 

120  .47 

120  .98 
121  .49 
122  .01 

'22  .53 

2  08088 
2.08273 
2.08457 
2  08641 
2  08824 

55 
56 

11 

59 

47  -46 
47  -79 
48  .11 
48  .41 
48  .76 

1.67636 
1.67930 
1.68223 
1.68515 
1.68806 

68  .73 

69  .12 
69  .51 
69  .90 
70  .29 

1.83716 
1.83960 
1.84204 
1.84447 
1.84690 

93  .92 
94  -38 
94  -83 
95  -29 
95  -74 

1.97279 
1.97488 
1.97697 
1.97905 
1.08112 

123  .05 

123  .57 

124  .09 

124  .61 

125  .13 

2.OOOO7 
2  09190 
2.OQ372 
2  09554 
2  09735 

60 

49  -°9 

1.69096 

70.  68 

1.84031 

96  .20 

1.98320 

T25  .65 

2.09917 

636 


TABLE    VI Ii   A. 


2  sin2  \t 

—     ; 77 — . 

sin  i" 


8 

I 

91 

i 

IO 

m 

ii 

m 

•m 

log  m 

m 

log  m 

•m 

log  m 

m 

log  m 

0 

I 

2 

3 
4 

126  .17 
126  .70 

127  .22 

1-27  -75 

2.09917 
2.10098 

2.  10278 
2.10458 
2.10637 

I59".o2 

160  .20 
160  .80 
161  .39 

2.20146 
2.20307 
2.20467 
2.20627 
2.20787 

I96".32 

196  .97 
197  .63 
198  .28 
198  .94 

2.29296 
2.29441 

2  29586 
2.29730 
2.29874 

237"-54 
238  .26 
238  .98 
239  .70 
240  .42 

2-37574 
2-37705 
2.37836 
2.37967 

2  38098 

7 
8 

128  .28 
128  .81 

129  .34 
129  .87 
130  .40 

2.10817 
2.10995 
2.III74 
2.H352 
2.II530 

161  .98 
162  .58 
163  .17 
'63  -77 
164  .37 

2.20946 
2.21106 
2.21264 
2.21423 
2.21581 

199  .60 

200  .26 
2OO  .92 
201   .59 
202  .25 

2.30017 
2.30l6l 
2.30304 
2.30447 
2.30590 

241  .14 
241  .87 
242  .60 
243  -33 
244  .06 

2.38229 
2.38360 
2.38490 
2.38619 

2.38749 

10 
ii 

130  .94 

i3r  -47 

2.II707 
2.11884 

164  .97 
Jjg  '57 

2.21739 
2.21897 

202  .92 

203  .58 

2.30732 
2.30874 

244  .79 
245  -52 

2.38879 
2.39009 

!3 

14 

132  -55 
133  -°9 

2.12237 
2  12413 

166  .77 
167  -37 

2.22212 
2.22369 

204  .92 
205  .59 

2.31158 
2.31300 

246  .98 
247  .72 

2  39267 
2.39396 

IS 

17 
18 

133  -63 

134  -71 
135  -25 
135  .80 

2.12589 
2.12764 
2.12939 

2.13288 

167  .97 
168  .58 
169  .19 
169  .80 
170  .41 

2.22525 
2.22682 
2.22838 
2.22994 
2.23150 

206  .26 
206  .93 
2O7  .60 
208  .27 
208  .94 

2.31441 
2.31582 
2.31723 
2.31864 
2.32004 

248  .45 
249  .19 
249  .93 
250  .67 

2.39525 
2.39654 
2.39782 
2.3991© 
2.40038 

20 
21 
22 

23 
24 

I-36  -34 
136  .88 
137  -43 
137  -98 
138  -53 

2.13462 

2.13635 
2.13809 
2.13982 
2.14154 

171  .02 
171  .63 
172  .24 
172  .85 

*73  -47 

2.23304 
2.23459 
2.23614 
2.23768 
2.23922 

209  .62 
2IO  .30 
210  .98 

211  .66 

212  .34 

2.32144 
2.32284 
.2.32424 
2.32563 
2.32703 

252  .15 

252  .89 
253  -63 
254  -37 

255  -12 

2.40l66 
2.40294 
2.40421 
2.40548 
2.40675 

25 
26 

11 

29 

139  .08 
139  .63 
140  .18 
140  .74 
141  .29 

2.14326 
2.14498 
2.14670 
2.14841 
2.I50II 

174  .08 
174  .70 

175  .32 
*75  -94 
176  -56 

2.24076 
2.24230 
2.24383 
2.24536 
2.24689 

213  .02 
.213  .70 
2I4  .38 
215  .07 

215  -75 

2.32842 
2  32980 
2.33119 
2.33258 
2.33396 

255  -87 
256  .62 

257  -37 
258  .12 

258  .87 

2.40802 
2.40929 
2.4,055 
2.4Il8l 
2.41307 

30 
31 
32 
33 

34 

141  -85 
142  .40 
142  .96 
!43  -52 
144  .08 

2.I5I82 
2.15352 
2.15522 
2.15691 
2.15860 

177  .18 
177  .80 
178  .43 
179  -05 
179  .68 

2.24842 
2.24994 
2.25146 
2.25297 
2.25449 

216  .44 

217  .12 

217  .81 

218  .50 

219  .19 

2-33534 
2.33671 
2.33809 

2.34083 

259  .62 
260  .37 

26l  .12 

261  .88 
262  .64 

2-4*434 
2.4,560 
2.41685 
2.4,8!, 
2.41936 

35 
36 

144  .64 

145  .20 
MS  .76 
H6  -33 

2.16029 
2.16198 
2.16366 
2.16534 
2.16701 

180  .30 
180  .93 
181  .56 
182  .19 
182  .82 

2.25600 
2.25751 
2.25902 
2.26052 
2.26202 

219  .88 

220  .58 
221  .27 
221   .97 
222  .66 

2.34220 

2-34357 
2-34493 
2.34630 
2.34766 

263  .39 
264  .15 
264  .91 
265  .68 
266  .44 

2.42061 
2.42,86 
2.42310 
2.42435 

4° 
41 

42 
43 
44 

I47  .46 
148  .03 
148  .00 
149  .17 
149  .74 

2.16868 
2.17035 
2.17202 
2.17368 
2-17534 

183  .46 
184  .09 
184  .72 
185  -35 
185  .99 

2.26352 
2.26501 
2.26651 
2.26800 
2.26949 

223  .36 
224  .06 
224  .76 
225  .46 
226  .16 

2.34901 
2.35037 
2.35172 
2.35307 
2.35442 

267  .20 
267  .96 
268  .73 
269  .49 
270  .26 

2.42683 
2.42807 
2.42931 

2.43055 
2.43,78 

45 
46 
47 
48 
49 

150  .31 

150  .88 
!5i  -45 
152  .03 
152  .61 

2.17700 
2.17865 
2  18030 
2.18194 
2.18359 

186  .63 
187  .27 
187  .91 
188  .55 
189  .19 

2.27097 
2.27246 
2.27394 
2.27542 
2.27689 

226  .86 
227  -57 
228  .27 
228  .98 
229  .68 

2-35577 
2.35712 
2.35846 
2.35980 

2  36114 

271  .02 
27I  .79 
272  .56 

273  -34 

274  .11 

2.43302 
2.43425 
2.43548 
2.43670 
2-43793 

50 

52 
53 
54 

153  -19 
J53  -77 
J54  -35 
154  -93 
X55  -51 

2.18523 
2.18687 
2.18850 
2.IOOI3 
2.19176 

189  .83 
190  .47 

191  .12 
191  .76 
192  .41 

2.27836 
2.27984 
2.28130 
2.28277 
2.28423 

230  .39 
231  .10 
231  .81 
232  .52 
233  -24 

2.36248 

2-k='5 
2.^6648 
2.36781 

274  .88 
275  -65 
276  -43 

277  .20 
277  -98 

2.44037 
2.44159 
2.44281 
2.44403 

ii 

57 
58 
59 

156  .09 
156  .67 
157  -25 
'57  -84 
158  -43 

2.19338 
2.I95OO 
2.19662 
2.19824 
2.19985 

193  .06 

193  -71 
194  .36 

195  •<! 

195  -66 

2.28569 
2.28715 
2.2886l 
2.29006 
2'.29I5I 

233  -95 
234  .67 
235  -38 
236  .10 
236  .82 

2.36913 
2.37046 
2.37178 
2.37310 
2.37442 

278  .76 

279  -55 
280  .33 

28l  .12 
28  1  .90 

2-44525 
2.44646 
2.44767 
2.44888 
2.45009 

60 

I^Q  .02 

7.20146 

106  .32 

2.20206 

237  -54 

2  37574 

282  68 

2.45130 

TABLE    VJJj  A. 


637 


2  sin2  \t 
sin  i" 


r  — 

12 

m 

13 

m 

14 

• 

15 

m 

nt 

log  m 

M 

log  m 

m 

log  nt 

M 

logm 

0 

I 

2 

3 

4 

2&2f'.68 

283  .47 
284  .26 
285  .04 
285  .83 

2.45130 
2.45250 

2-45371 
2.45491 
2.45611 

331"-74 
332  -59 
333  -44 
334  -29 

335  -i5 

2.52081 
2.52192 
2.52303 
2.52414 
2.52525 

384"-  74 
385  .65 
386  .56 
387  .48 
388  .40 

2.58516 
2.58619 
2.58722 
2.58825 
2.58928 

44i"-63 
442  .62 
443  -60 
444  -58 
445  -56 

2.64506 
2.64603 
2.64699 
2.64795 
2.64891 

1 

9 

286  .62 
287  .41 

288  .20 

289  .00 

289  .79 

2-45731 
2.45850 
2.45970 
2.46089 
2.46209 

336  .00 
336  -86 
337  -72 
338  -58 
339  -44 

2-52635 

2  52746 
2.52856 
2.52907 
2.53077 

389  -32 
390  .24 
391  .16 

392  .09 
393  -OI 

2.59031 
2-59*34 
2.59236 

2-59339 
2.59441 

446  -55 
447  -54 
448  -53 
449  -51 
450  .50 

2.64987 
2.65083 
2.65179 
2.65274 
2.65370 

10 

ii 

12 

13 

*4 

290  .58 
291  .38 
292  .18 
292  .98 

293  -78 

2.46328 
2.46446 
2.46565 
2.46684 
2.46802 

340  -3° 
341  .16 
342  .02 
342  .88 

343  -75 

2-53l87 
2.53297 
2.53406 
2.535i6 
2.53625 

393  -94 
394  -86 
395  -79 
396  .72 
397  65 

2-59543 
2  59645 
2-59747 
2.59849 
2  59951 

45i  -50 
452  -49 
453  -48 
454  .48 
455  -47 

2.65466 
2.65561 
2.65656 
2-65751 
2.65846 

15 
16 

\l 

T9 

294  .58 

295  -38 

296  .18 
296  .99 
297  .79 

2.46920 
2.47038 
2.47156 

2.47274 
2.47392 

344  62 
345  -49 
346  -36 
347  -23 
348  .10 

2-53735 
2.53844 
2-53953 
2.54062 
2.54170 

398  .58 
399  -52 
400  .45 
401  -38 
402  .32 

2.60052 
2.60154 
2.60255 
2.60357 
2.60458 

456  -47 
457  -47 
458  -47 
459  -47 
460  .47 

2.65941 
2.66036 
2.66131 
2.66225 
2.66320 

20 
21 
22 
23 
24 

298  .60 
299  .40 

3OO  .21 
301  .02 

301  .83 

2.47509 
2.47626 

2-47743 
2.47860 

2-47977 

348  -97 
349  -84 
350  -71 
35i  -58 
352  .46 

2.54279 
2.54387 
2.54496 
2.54604 
2.54712 

403  .26 
404  .20 
405  .14 
406  .08 

407  .02 

2.60559 
2.60660 
2.60760 
2.60861 
2.60961 

461  .47 
462  .48 
463  .48 
464  .48 
465  .49 

2.66414 
2.66509 
2.66603 
2.66697 
2.66791 

25 
26 
27 
28 
29 

302  .64 
303  -46 

3°4  -27 

305  .09 

3°5  -9° 

2.48094 
2.48210 
2.48327 
2.48443 
2-48559 

353  -34 
354  -22 
355  -io 
355  -98 
356  .86 

2.54820 
2.54928 
2.55035 
2  55M3 
2-55250 

407  .96 
408  .90 
409  .84 
410  .79 

411  -73 

2.61062 
2.6x162 
2.61263 
2.61363 
2.61463 

466  .50 
467  -5l 
468  .52 
469  -53 
47°  54 

2.66885 
2.66979 
2.67073 
2.67166 
2.67260 

3° 
3i 
32 
33 
34 

306  .72 
3°7  -54 
308  .36 
309  -18 
310  .00 

2.48675 
2.48790 
2.48906 
2.49021 

2  49136 

357  -74 
358  .62 

359  -5i 

360  .39 
361  .28 

2.55358 
2-55465 
2.55572 

2  55679 

2-5578S 

412  .68 
4i3  -63 
4H  -59 
4i5  -54 
416  .49 

2.61563 
2.61662 
2.61762 
2.61861 
2.61961 

47'  -55 
472  -57 
473  -58 
474  .60 
475  -62 

2-67353 
2.67446 

2-67539 
2.67633 
2.67726 

P 

37 
38 
39 

310  .82 
311  -65 
3"  -47 
3(3  -30 
314  .12 

2.49251 
2.49366 
2.49481 
2.49596 
2.49711 

362  .17 

363  -07 
363  .96 
364  -85 
365  -75 

2.5,892 
2.5^999 
2.56105 
2.56211 
2.56317 

417  .44 
418  .40 
4J9  -35 
420  .31 
421  .27 

2.62060 
2.62159 
2.62258 
2.62357 
2.62456 

476  .64 
477  -65 
478  .67 

479  -7° 
480  .72 

2.67818 
2.67911 
2.68004 
2.68097 
2.68189 

40 
4i 

42 
43 
44 

3M  -95 
315  .78 
316  .61 
3*7  -44 
318  .27 

2.49825 

2-49939 
2.50053 
2.50167 
2.50281 

366  .64 
367  -53 
368  .42 
369  .31 

370  .2T 

2.56423 
2.56529 
2.56635 
2.56740 
2.^6846 

422  .23 
423  -19 
424  -15 
425  -ii 
426  .07 

2-62555 
2.62654 
2.62752 
2.62850 
2.62949 

48  c  .74 
482  .77 

483  -79 
484  .82 

485  -85 

2.68281 

2  68374 
2.68466 
2.68558 
2.68650 

45 
46 
47 
48 
49 

319  .10 
3i9  -94 
320  .78 
321  .62 
322  .45 

2.50394 
2.50508 
2.50621 
2-50734 
2.50847 

37i  •" 
372  .01 
372  .91 
373  -82 
374  -72 

2.56951 
2.57056 
2.57161 
2.57266 
2.57371 

427  -04 
428  .01 
428  .97 
429  -93 
430  .90 

2.63047 

2  .63  J  45 
2.63243 
2.63341 
2.63438 

486  .88 
487  .91 
488  .94 
489  -97 
491  .01 

2.68742 
2  68834 

2.68926  ! 

2.6(5017 

2  ^QIOQ 

.50 
51 
52 
53 
54 

323  .29 
324  -13 
324  -97 
325  -81 
326  .66 

2.50960 
2.51073 
2.51185 
2.51298 
2.51410 

375  -62 
376  .52 
377  -43 
378  -34 
379  -26 

2.57476 
2.57580 
2.57685 
2.57789 
2.57893 

43i  -87 
432  -84 
433  -82 
434  -79 
435  -76 

2.63536 
2.63634 
2.63731 
2.63828 
2.63925 

492  -05 
493  -08 
494  .12 
495  .'5 

4q6  .10 

2.69201 
2.69292 
2.69383 
2  69474 
2.69565 

55 
56 

57 
58 
59 

327  -50 
328  .35 
329  -19 
330  -04 
33°  .89 

2.51522 

2.51634 
2.51746 
2.51858 
2.51969 

380  .17 
381  .08 
381  .99 
382  .90 
383  .82 

2-57997 
2.58101 
2.58205 
2.58309 
2.58412 

436  -73 
437  -7i 
438  .69 
439  -67 
440  .6s 

2.64022 
2.64119 

2  64216 
2.64313 
2.64410 

497  -23 
498  -28 
499  -32 
SOD  .37 
501  -4T 

2.69656 
2.69747 
2.60838 
9  60029 
2  70019 

60 

331  74 

2.52081 

384  .74 

2.58516 

441  .63 

2.64506 

502  .46 

2  7OTO9 

TABLE    VIII  A. 


2  sin2  \t 

nt  =  — ; — -. 

sm  i" 


J 

i 

6m 

ll 

m 

Ii 

m 

IS 

m 

m 

log  »/ 

m 

log  m 

M 

log  nt 

m 

log  nt 

o 

I 

2 

3 
4 

502".s 
503  -5 
504  -5 
5°5  -6 
506  .6 

2.70109 
2.70200 
2.70291 
2.70381 
2.70471 

567"-  2 
568  .3 

569  -4 
570  -5 
57t  -6 

2-75373 
2  75458 
2-75H3 
2.75628 
2-757I3 

635"-9 
637  -o 
638  .2 

639  -4 
640  .6 

2.80336 
2.80416 
2.80496 
2.80576 
2.80656 

7o8//.4 
709  .7 
710  .9 
712  .1 
7i3  -4 

2.85029 
2.85105 
2.85181 
2.85257 
2-85333 

i 

7 
8 

507  -7 
508  .8 
509  -8 
510  .9 

5"  -9 

2.70561 
2.70651 
2.70741 
2.70830 
2.70920 

572  -8 
573  -9 
575  -o 
576  -i 
577  -2 

2.75798 
2.75883 
2.75967 
2.76052 
2.76136 

641  .7 
642  .9 
644  .1 
645  -3 
646  .5 

2.80736 
2.80816 
2.80896 
2.80976 
2.81056 

714  .6 
7J5  -9 
717  .1 
718  .4 
719  .6 

2-85409 
2.85485 
2.85561 
2.85636 
2.85712 

10 
ii 

12 
13 
14 

5J3  -° 
514  .0 
5*5  -I 
516  .1 
5*7  -2 

2.7IOIO 
2.71099 
2.71188 
2.71278 
2.71367 

578  -4 
579  -5 
580  .6 
58i  .7 
582  .  .9 

2.76220 
2.76304 
2.76388 
2.76472 
2.76556 

647  .7 
648  .9 
650  .0 

65I  .2 

652  .4 

2.81135 
2.81215 
2.81295 

2-81375 
2.81454 

720  .9 
722  .1 
723  -4 
724  .6 
725  -9 

2.85787 

2  85863 
2.85938 
2.86014 
2.86089 

:t 

17 

18 
19 

5i8  -3 
519  -3 
520  .4 
52i  .5 
522  .5 

2.71456 
2.71545 
2.71634 
2.71723 
2.71811 

584  -o 
585  -i 

586  .2 

587  -4 
588  .5 

2.76640 
2.76724 
2.76808 
2.76892 
2.76976 

653  -6 
654  -8 
656  .0 

657  .2 
658  .4 

2.81533 
2.81612 
2.81691 
2.81770 
2.81849 

727  .2 
728  .4 
729  .7 
730  -9 

732  .2 

2.86164 
2.86239 
2.86314 
2.86389 
2.86464 

20 
21 
22 

23 
24 

523  -6 
524  -7 
525  -7 
526  .8 
527  -9 

2.71900 
2.71989 
2.72077 
2.72165 
2.72254 

589  .6 
590  .8 
59i  -9 
593  -o 
594  -2 

2.77059 
2.77143 
2.77226 
2.77309 
2.77392 

659  -6 
660  .3 
662  .0 

663  .2 
664  .4 

2.81928 
2.82007 
2.82086 
2.82165 
2.82244 

733  -5 
734  -7 
736  -o 
737  -3 
738  -5 

2.86539 
2.86614 
2  86689 
2.86763 
2.86838 

2| 
26 

\  27 
28 
29 

529  .0 
53°  -o 
53i  -i 
S32  .2 
533  -3 

2.72342 
2.72430 
2.72518 
2.72606 
2.72694 

595  -3 
596  -5 
597  -6 
598  -7 
599  -9 

2.77476 
2-77559 
2.77642 
2.77724 
2.77807 

665  .6 
666  .8 
668  .0 
669  .2 
670  .4 

2.82322 
2.82401 
2.82479 
2.82558 
2.82636 

739  -8 
741  .1 

742  -3 
743  -6 
744  -9 

2.86912 
2.86987 
2.87061 
2.87136 
2.872IO 

3° 
31 
32 

33 
34 

534  -3 
535  -4 
536  -5 
537  -6 
538  -7 

2.72781 
2.72869 
2.72957 
2.73044 
2.73132 

601  .0 

602  .2 

603  .3 

604  .5 
605  .6 

2.77890 
2-77973 
2.78056 
2.78138 
2.78220 

671  .6 
672  .8 
674  -1 
675  -3 
676  .5 

2.82714 
2.82792 
2.82870 
2.82948 
2.83026 

746  .2 

747  -4 
748  .7 
750  .0 
75'  -3 

2.87284 
2.87358 
2.87432 
2.87506 
2.87580 

35 
36 

1 

539  -7 
540  .8 

54i  -9 
543  -o 
544  -i 

2.73219 
2.73306 

2-73393 
2.73480 

2-73S67 

606  .8 
607  .9 
609  .1 
610  .2 
611  .4 

2.78302 
2.78385 
2.78467 
2.78549 
2.78631 

677  -7 
678  .9 
680  .1 
681  .3 
682  .6 

2.83104 
2.83182 
2.83260 
2-83337 
2.83414 

752  -6 
753  -8 
755  -i 
756  -4 
757  -7 

2.87654 
2  87728 
2  87802 
2.87876 
2.87949 

40 
4i 
42 
43 
44 

545  -2 
546  -3 
547  -4 
548  -5 
549  -5 

2-73654 
2-73741 
2.73827 
2,73914 
2.74001 

612  .5 

613  -7 
614  .8 
616  .0 

617  .2 

2.78713 
2.78795 
2.78877 
2.78958 
2.79040 

683  .8 
685  .0 
686  .2 
687  .4 
688  .7 

2.83492 

2.  831570 
2.83648 
2.83725 
2.83802 

759  -o 

760  .2 

76i  -5 
762  .8 
764  -1 

2.88023 
2.88096 
2.88170 
2.88243 
2.88317 

45 
46 

3 

49 

550  .6 
55i  -7 
552  -8 
553  -9 
555  -o 

2.74087 
2.74173 
2.74259 
2.74346 
2.74432 

618  .3 
619  .5 
620  .6 
621  .8 
623  .0 

2.79121 
2.79203 
2.79284 
2.79366 
2.79447 

689  .9 
691  .1 
692  .4 
693  -6 
694  -8 

2-83879 
2.83957 
2.84034 
2.84III 
2.84188 

765  -4 
766  .7 
768  .0 
769  -3 
770  .6 

2.88^90 
2.88463 
2.88536 
2.88filO 

2.886j83 

50 

5i 
52 
53 
54 

556  -i 
557  -2 
558  -3 
559  -4 
560  .5 

2-74518 
2.74604 
2.74690 
2-74775 
2.74861 

624  .1 
625  -3 
626  .5 
627  .6 
628  .8 

2.79528 
2.79609 
2.79690 
2.79771 
2.79852 

696  .0 

697  -3 
698  .5 
699  .7 
701  .0 

2.84264 
2.84341 
2.844l8 
2.84495 
2.84571 

771  .9 
773  -i 

774  -5 
775  -7 
777  .1 

2.88756 
2.88828 
2.88901 
2.88974 
2.89047 

It 

3 

59 

561  -7 
562  .8 
563  -9 

%~ 

2.74947 
2.75032 
2.75118 
2.75203 
2.75288 

630  .0 

631  .2 
632  .3 

633  -5 
634  -7 

2-79933 
2.80014 
2.80094 
2.80175 
2.80255 

7O2  .2 

703  -5 
704  .7 

7°5  -9 
707  .1 

2.84648 
2.84724 
2.84801 
2.84877 
2.84953 

778  .4 
779  -7 
781  .0 
782  .3 
783  .6 

2.89119 
2.89192 
2.89265 
2-89337 
2.80409 

60 

567  .2 

2-7537? 

635  -9 

2.80336 

708  .4 

2  85029 

784  -9 

2.89481 

TABLE    VIII  A. 


639 


2  sin'  jt 
sin  i" 


20 

n 

21 

. 

22 

m 

23 

m 

m 

log  nt 

M 

log  nt 

m 

log  m 

nt 

log  m 

o 

I 

2 

3 

4 

784".  9 

786  .2 

787  -5 
788  .8 
79°  -1 

2.89481 
2.89554 
2.89626 
2.89698 
2.89770 

865".3 
866  .6 
868  .0 
869  .4 
870  .8 

2.93717 
2.93786 

2-93855 
2.93923 
2.93992 

949".6 
951  .0 
952  -4 
953  -8 
955  -3 

2-97755 
2.07820 
2.97886 
2.97952 
2.98017 

1037".  8 
1039  -3 
1040  .8 
1042  .3 
1043  .8 

3  01613 
3-ol675 
3-01738 
3  01801 
3.01864 

i 

7 
8 
9 

791  .4 
792  .7 
794  -o 
795  -4 
796  .7 

2.89842 
2.89914 
2.89986 
2.90058 
2.90130 

872  .1 
873  -5 
874  -9 
876  .3 
877  -6 

2.94061 
2.94129 
2.94198 
2.94266 
2-94335 

956  -7 
958  .2 
959  -6 
961  .1 
962  .5 

2.98083 
2.98148 
2.98214 
2.98279 
2.98344 

1045  .3 
1046  .8 
1048  .3 
1049  -8 
1051  -3 

3.01926 
3.01989 
3.02052 
3.02114 
3.02177 

10 

IT 

12 
13 
14 

798  .0 
799  -3 
800  .7 
802  .0 
803  .3 

2.90202 

2  90274 
2.90346 
2.90417 
2.90489 

879  .0 
880  .4 
88  1  .8 
883  .2 
884  .6 

2.94403 
2.94471 
2.94540 
2.94608 
2.94676 

963  -9 
965  -4 
966  .9 
968  .3 
969  .8 

2.98410 
2.98475 
2.98540 
2.98605 
2.98670 

1052  .8 
1054  .3 
1055  -9 
1057  -4 
1058  .9 

3.02239 
3.02302 
3.02364 
3.02426 
3.02489 

16 

*7 

18 

19 

804  .6 
806  .0 
807  .3 
808  .6 
809  .9 

2.90560 
2.90632 
2.90703 
2.90774 
2.90845 

886  .0 
887  .4 
888  .8 

890  .2 

891  .6 

2.94744 

2.94812 
2.94880 
2.94948 
2.95016 

971  .2 
972  -7 

974  -1 
975  -5 
977  -o 

2-98735 
2.98800 
2.98865 
2.98930 
2.98995 

1060  .4 
1062  .0 
1063  .5 
1065  .0 

1066  .5 

3-02551 
3.02613 
3.02675 
3-02737 
3.02799^,' 

20 

21 
22 
23 
24 

8n  -3 
812  .6 
813  -9 
8.5  .2 
816  .6 

2.90917 
2.90988 
2.91058 
2.91129 
2.91200 

893  .0 
894  .4 
895  -8 

897  .2 

898  .6 

2.95084 
2.95152 
2.95219 
2.95287 
2-95355 

978  -5 
979  -9 
98,  .4 
982  .9 
984  .4 

2.99060 
2.99125 
2.99189 
2.99254 
2.99319 

1068  .1 
1069  .6 
1071  .1 
1072  .6 

1074'  .2 

3.02861 
3-02923  , 
3-02985  \ 
3-03047 
3  03109 

25 
26 
27 
28 
29 

817  .9 

819  .2 
820  .5 
821  .9 
823  .2 

2.9I27I 
2.91342 
2.91413 
2.91484 
2.9I555 

900  .0 
901  .4 

902  .8 

904  .2 

905  .6 

2.95422 
2.95490 

2-95557 
2.95625 
2.95692 

985  -8 
987  -3 
988  .8 
990  -3 
991  .8 

2.99383 
2.99448 
2.9*5512 
2.99576 
2.99641 

*075  -7 

1077  .2 

1078  .7 
1080  .3 

1081  .8 

3-031?1 
3-03232 
3-03294 
3-03356 
3-034*7 

3° 
31 

824  .6 

825  .9 

2.91625 
2.91696 

goi  .0 
908  .4 

2-95759 
2.95827 

993  -2 

994  -7 

2.99705 
2.99769 

1083  .3 
1084  .8 
1086  .4 

3-03479 

3-03540 

33 
34 

828  .6 
829  .9 

2.91837 
2.91907 

9II  .2 

912  .6 

2.95961 
2.96028 

997  .6 
999  -1 

2.99898 
2.99962 

1087  .9 
1089  .5 

3.03663 
3-03725 

35 
36 
37 
38 
39 

831  .2 

832  .6 
833  -9 

H3I  'I 
836  .6 

2.91977 
2  92048 
2.92118 
2.92188 
2.92258 

914  .0 
915  -5 
916  .9 
918  .3 
919  .7 

2.96095 
2.96162 
2.96229 
2.96296 
2.96362 

looo  .6 

1002  .1 

1003  .5 

1005  .0 
1006  .5 

3.00026 
3.00090 
3.00154 
3.00218 
3.00282 

1091  .0 
1092  .6 
1094  .1 

i°95  -7 
1097  .2 

3-03787 
3.03848 
3.03909 
3-03970 
3  04031 

40 
41 
42 
4? 
44 

838  .0 

839  -3 
840  .7 
842  .0 
843  -4 

2.92328 
2.92398 
2  92468 
2.92538 
2.92608 

921  .1 
922  .5 
923  -9 
925  -3 
926  .8 

2.96429 
2.96496 
2.96563 
2.96630 
2.96696 

1008  .0 

1009  .4 

1010  .9 

IOI2  .4 
1013  .9 

3.00746 
3.00409 
3-00473 
3-00537 
3.00600 

1098  .8 
noo  .3 
noi  .9 

1103  .4 

1  105  .0 

3.04092 

3-°4I53 
3.04214 
3.04275 
3-04336 

9. 

47 
48 
49 

844  -7 
846  .1 

847  -5 
848  .9 

850  .2 

2.92677 
2.92747 
2.92817 
2.92886 
2.92956 

928  .2 
929  .6 
931  .0 

932  -4 
933  -8 

2  96763 
2  96829 
2.96896 
2.96962 
2  97028 

1015  .4 

1016  .9 
1018  .4 
1019  .9 

1021  .4 

3.00664 
3.00728 
3.00791 
3-00855 
3.00918 

1106  .5 
1108  .1 
1109  .6 

IIII  .2 
III2  .7 

3-04397 
3-04458 
3.04519 
3-04580 
3  0464! 

So 
51 
52 
53 
54 

851  .6 

852  .9 
854  -3 
855  -7 
857  -i 

2.93026 
2.93096 
2.93164 
2-93233 

2-933°3 

935  •* 

936  .6 
938  .1 
939  -5 
94°  -9 

2.97095 
2.97161 
2.97227 
2.97293 

2  97359 

1022  .8 
1024  .3 

1025  .8 
1027  3 
1028  .8 

3.00981 
3.01045 
3.01108 
3.01171 
3-01234 

1114  .3 

1115  -8 
1117  .4 
1118  .9 

"20  .5 

3.04701 
3-04762 
3-04823 
3.04883 
3.04944 

55 
56 
57 
58 

<n 

858  .4 
8<;9  .8 
86z  .1 
'  862  .5 
863  .9 

2-93372 
2.93441 
2  935.10 
2-93579 
2.93648 

942  -3 
943  -8 
945  .2 
946  .6 

2  97425 
2.97491 
2-97^57 
2  97623 

1030  .3 
1031  .8 

io33  -3 
1034  .8 
1036  .3 

3.01298 
3.01361 
3  01424 
3.01487 
3-OI550 

1122  .O 

1123  .6 
1125  .1 
1126  .7 

IT28  .3 

3.05004 
3.0506;; 
3-05125 
3-°S*8s 
3.05246 

i   60 

865  -3 

2.93717 

949  -6/ 

2-97755 

1037  8 

3.01613 

1129  .9 

3.05306  j 

640 


TABLE    VIII  A. 


2  sin2  \t 


0 

24 

m 

2_ 

>m 

2 

3m 

2 

?m 

m 

log  m 

m 

log  m 

M 

log  m 

m 

log  m 

0 
2 

3 

4 

ii29".9 
"3l  -4 
1133  .0 
1134  .6 
1136  .2 

3.05306 
3.05366 
3.05426 
3.05487 
3-05547 

I225".9 
1227  .5 

1229  .2 

1230  .8 
1232  .5 

3.08848 
3.08906 
3.08964 
3.09022 
3.09079 

i325"-9 
1327  .6 
1329  3 
1331  -o 
J332  -7 

3.12252 
3.12307 
3-12363 
3.12418 
3.12474 

i429'/.7 
i43i  -4 

1433  -2 

*435  -o 
MS6  -7 

3-I5526 
3-  T  558o 
3.I5033 

3.15686 
3-J574o 

5 
6 
7 
8 
9 

1137  .8 
"39  -3 
1140  .9 
1142  -5 
1144  .0 

3.05607 
3.05667 
3-°5727 
3-05787 
3-05847 

1234  .1 
1235  -7 
I237  -3 
1239  -o 
1240  .6 

3-09i37 
3-°9'95 
3.09252 
3.09310 
3-09367 

X334  -4 
1336  .1 
1337  -8 
'339  -5 
1341  .2 

3.12529 
3-12585 
3.12640 
3.12695 
3-12751 

H38  -5 
1440  .3 
1442  .1 
1443  -9 
J445  6 

3-  '5793 
3-15847 
3.15900 

3-15953 
3.16007 

TO 
II 
12 
13 

M 

1145  .6 

1147  .2 

1148  .8 
1150  .4 
1152  .0 

3-°59°7 
3.05966 
3.06026 
3.06086 
3.06146 

1242  .3 
1243  .9 
1245  .6 

1247  .2 
1248  .9 

3-09425 
3.09482 
3  09540 
3-09597 
3-09655 

1342  .9 
1344  .6 
1346  -3 
1348  .0 
1349  -7 

3.12806 
3.12861 
3.12916 
3.12971 
3-13026 

M47  -4 
1449  .2 
1451  .0 
1452  .8 
J454  -5 

3.16060 
3.16113 
3.16166 
3.16220 
3.16273 

15 
16 
*7 

18 
i9 

"53  -6 
1155  .2 
1156  .8 
1158  .3 
1159  -9 

3.06205 
306265 
3.06324 
3.06384 
3.06444 

1250  .5 
1252  .2 

1253  -8 
I255  -5 
1257  .1 

3.09712 
3.09769 
3  09826 
3.09883 
3.09941 

T35l  -4 

1353   •» 

*354  -9 
1356  .6 
J358  .3 

3.13081 
3-i3'36 
3-*3l9l 
3-13246 
3-I1301 

*456  -3 
1458  .1 
M59  -9 
1461  .6 
1463  .4 

3.  16326 
3-16379 
3.16432 
3.16485 
3-16538 

20 
21 
22 

23 
24 

1161  .5 
1163  .1 
1164  .7 
1166  .3 
1167  .9 

3.06503 
3.06562 
3.06622 
3.06681 
3.06740 

1258  .8 
1260  .5 

1262  .2 

1263  .8 
1265  .5 

3.09998 
3-I0o55 
3.10112 
3.10169 
3.10226 

1360  .1 
1361  .8 

i3fi3  -5 

1365  .2 

1367  .0 

3-J3356 
3-I34" 
3.13466 
3-I352i 
3-I3576 

1465  .2 
1466  9 
1468  .7 
T47°  -5 
1472  .3 

3-  '6591 
3.16643 
3.16696 
3.16749 
3.16802 

25 
26 
27 
28 
29 

1169  .5 
1171  .1 
1172  .7 
"74  -3 
"75  -9 

3.06800 
3.06859 
3.0691$ 
3.06977 
3.07036 

1267  .1 
1268  .8 
1270  .5 
1272  .1 
'273  -7 

3.10283 
3.10340 
3.10396 
3-10453 
3.10510 

1368  7 
1370  .4 
1372  .1 
1373  -9 
I:!75  -6 

3-  *363* 
3.13686 
3-I3740 
3-13795 

3-  1  38  so 

1474  .1 
M75  -9 
J477  -7 
T479  -5 
1481  .3 

3-16855 
3  16907 
3.10900 

3-I7°n 
3.17066 

30 

31 
32 
33 
34 

"77  -5 
"79  -1 
1180  .7 
1182  .3 
1183  9 

3.07095 

3-°7T54 
3.07213 
3.07272 
3-07331 

1275  -4 
1277  .1 
1278  .8 
1280  .4 
1*82  .1 

3-  '0567 
3  10623 
3.10680 
3-  J0737 
3.10703 

'377  -3 
1379  -o 
1380  .8 
1382  .5 
1384  .2 

3.13904 
3-  '3959 

:-i4=>T3 
3.74068 

3.14122 

1483  -i 
1484  .9 
,486  .7 
1488  .5 
1490  3 

3.17.18 

3.17170 
3.17223 
3-17275 

3-!7327 

35 
36 

% 

39 

"85  -5 
1187  .1 
i  i  88  .7 
1190  .3 
1191  .9 

3-07389 
3  07448 
3-07507 
3.07566 
3.07625 

1283  .8 
1285  .5 
1287  .1 
1288  .8 
1290  .5 

3.10850 
3.10906 
3.10963 
3.11019 
3.11076 

1385  -9 
1387  -7 
1389  .4 

I39I  .2 

1392  .9 

3  H!77 
3.14231 
3-I428s 
3-I4>40 
3-T4^94 

1492  .1 
1493  -9 
'495  -7 
1497  -5 
1490  .3 

3.17380 
3-17431 

3  I74S5 
3-17538 
3  !759o  i 

4° 
41 
42 
43 
44 

"93  -5 
"95  -i 
1196  .7 
"98  -3 
"99  -9 

3  07683 
3.07742 
3.07801 
3.07859 
3.07918 

1292  .2 

1293  -8 
1295  -5 

1297  .2 

1298  .9 

3.11132 
3.11188 
3-  "245 
3.11301 
3-"357 

*394  -7 
1396  .4 
1398  .2 
1399  -9 
1401  .7 

3.14448 
3.14502 

3-14557 
3.14611 
3  ^665 

1501  .1 
1502  .9 
1504  -7 
1506  .5 
1508  .4 

3.17642 
3-17694 
3  *7746 

3-  '7799 
3-  '7851 

3 

47 
48 
49 

I2OI   .5 
1203  .1 
1204  .7 
1206  .4 
I2O8  .O 

3.07976 
308035 
3.08093 
3.08151 
3.08210 

1300  .5 

1302  .2 

1303  -9 
1305  .6 

*3°7  -3 

3  "4i3 
3.11469 
3-"525 
3.11  =  82 
3.11638 

1403  .4 

1405  .2 

1406  .9 
1408  .7 

^410  .4 

3-I47I9 
3-T4773 
3.14827 
3.1488: 
S-MQSS 

1510  .2 
1512  .O 

1513  .8 
1515  -6 
1517  -4 

3.17903 

3-17955 
3.18007 
3.  18059 
3.18111 

50 
5i 
52 
53 
54 

1209  .6 

1211  .2 
1212  .9 
1214  .5 
T2l6  .1 

3.08268 
3.08326 
3.08384 
3.08442 
3.o8=;oi 

1309  .0 
1310  .7 
1312  .4 
1314  .1 
mS  -7 

3.11694 
3  "750 
3.11805 
3.11861 
3."9i7 

I4I2  .2 

HT3  9 
14^5  -7 
1417  .4 

1419  .2 

3.14989 
3  15043 
3.15096 
S-^iSO 
3-I52°4 

1519  -2 
1521  .O 
I522  .9 

1524  -7 
1526  .5 

3.18163 
3.18215 
3.18267 
3  ^319 
3-18371 

$ 

11 

59 

1217  .7 
I2ig  .4 
1221  .O 
1222  .6 
1224  .2 

3.08559 
3.08617 
308675 
3-o8733 
3.08791 

i.3r7  -4 
1319  .1 
1320  .8 
1322  .5 
1324  .1 

3-  "973 
3.12029 
3.12085 
3.12140 
3.12196 

1420  .9 
1422  .7 
1424  .4 
1426  .2 
1427  .9 

3-'5258 
3-153" 
3-r5365 
3-I54I9 
3-I5472 

1528  .3 

1530  -2 

1532  -o 
1533  -8 

T535  -6 

3.1842- 
3.18474 
3.18-?-' 
3  '8578 
3.186-0 

60 

1225  .9 

3.08848 

1325  q 

^.  12252 

1429  7 

3-  T5526 

I537  -5 

7.18^8' 

^'ABLE    VIII  A. 


64: 


2  sin2  \t 


28 

• 

29 

m 

30 

m 

3i' 

n 

m 

log  m 

1H 

log  m 

m 

log  m 

m 

log  m 

o 

•2. 

3 
4 

i537"-5 
1539  -3 
1541  .1 
1542  .9 
1544  -8 

3.18681 
3-18733 
3.18784 
3.18836 
3.18887 

1649".! 
1651  .O 
1652  .9 

1654  -8 
1656  .7 

3-21725 

3  21775 
3.21825 
3.21875 
3.21924 

i  764".  6 
1766  .6 
1768  .5 
1770  -5 
1772  .5 

3.24665 

3-24713 
3.24761 
3.24810 
3.24858 

i884".o 
1886  .1 
1888  .1 
1890  .1 
1892  .1 

3.27509 
3.27556 
3.27602 
3.27649 
3.27695 

! 

9 

1546  .6 
15*8  .4 

1550  -2 
1552  .1 

1553  9 

3.18939 
3.18990 
3.19042 
3.19093 
3-t9i45 

1658  .6 
1660  .5 
1662  .4 
1664  .3 

1666  .2 

3  21974 
3.22024 
3.22073 
3.22123 
3.22172 

1774  -4 
1776  .4 
1778  .4 
1780  .3 
1782  -3 

3.24906 
3.24954 
3.25002 
3-25050 
3.25098 

1894  .2 
1896  .2 
l898  .2 
1900  .3 
1902  .3 

3.27742 
3.27788 
3.27835 
3.27881 

3  27928  i 

10 

ii 

12 

J3 
14 

1555  -8 
1557  -6 
*559  -5 
1561  -3 
1563  -2 

3.19196 
3-19247 
3-19299 
3-1935° 
3  19401 

1668  .1 

1670  .0 

1671  .9 

1673  -8 
1675  .7 

3.22222 
3.22272 
3-22321 

3.22371 
3.22420 

1784  -3 
1786  .2 

1788  .2 
I790  .2 
1792  .1 

3.25146 
3-25!94 
3.25242 
3.25289 
3  25337 

1904  -3 
1906  .4 
1908  .4 
1910  .4 
1912  .5 

3-27974 
3.28020 
3.28067 
3.28113 
3.28159 

17 
18 
19 

1565  -o 
1566  .9 
1568  .7 
1570  -5 
'572  -4 

3-'9452 
3-  '9  503 
3-19554 
3.19606 
3-I9657 

1677  .6 
1679  -5 
1681  .4 
1683  .3 

1685  .2 

3.22470 
3.22519 
3-22568 

?,.226l8 

3.22667 

1794  .1 
1796  .1 
1798  .1 

1800  .0 

1802  .O 

3-25385 
3-25433 
3.25480 
3-25528 
3-25576 

I9H  -5 
1916  .5 
1918  .6 
1920  .6 
1922  .7 

3.28206 
3.28252 
3.28298 
3-28344 
3  2839° 

20 

21 
22 

23 
24 

'574  -3 
1576  .1 
1578  .0 

1579  -8 
1581  .7 

3.19708 
3-I9759 
3.19810 
3.19861 
3.19912 

1687  .2 
1689  .1 
1691  .O 
1692  .9 

1694  .8 

3.22716 
3.22766 
3.22815 

3  22864 
3.22013 

1804  .0 
1806  .0 
1808  .0 

1809  .9 

1811  .9 

3.25624 
3-25671 
3257'9 
3.25766 
3.25814 

1924  .7 
1926  .8 
1928  .8 
1930  .9 
1932  .9 

3-28437 
3.28483 
3-28529 

3-28575 
3.28621 

25 

26 
27 
28 
29 

'583  -5 
1585  -3 
1587  .2 
1589  .1 
1590  .9 

3.19962 
3.20013 
3.20064 
3.20115 
3.20166 

1696  .7 
1698  .6 
1700  -5 
1702  -5 
1704  .4 

3.22963 
3.23012 
3.23061 
3.23110 
3-23?59 

1813  .9 
1815  .9 
1817  .9 
1819  .9 
1821  .9 

3.25862 
3-25909 
3-25957 
3.26004 
3-26051 

1934  -9 

1937  -0 

1939  .1 
1941  .1 

1943  -2 

3.28667 
3-28713 
3-28759 
3.28805 
3.28851 

30 

3l 

32 
33 
34 

1592  -7 
1594  .6 
1596  -5 
1598  -3 

l6<X>  .2 

3.20216 

3.20267 
3.20318 
3-20369 
3.20419 

1706  .3 

1708  .2 
17IO  .2 
1712  .1 

1714  .0 

3.23208 
3.23257 
3-23306 
3  23355 

3-23404 

5>  00  00  00  00 

bo  bo  bo  bo  oo 

3.26099 
3.26146 
3.26194 
3.26241 
3.26288 

1945  -2 
1947  -3 

1949  -3 
I95i  -4 
J953  -4 

3-28897 
3.28943 
3.28988 
3-29034 
3.29080 

35 
36 

1 

39 

I6O2   .1 
1604  .O 
1605  .9 
1607   .7 

1609  .6 

3.20470 
3.20520 
3.20571 
3.20621 
3.20672 

1715   .9 

1717  -o 
1719  .8 
1721  .7 
1723  .6 

3-23453 
3-23501 
^•23550 
3.23599 
3.23648 

00  00  00  00  00 

bo  bo  bo  bo  bo 

3.26336 
3.26383 
3.26430 
3.26477 
3.26524 

1955  -5 
'957  -6 
1959  .6 
1961  .7 
1963  -8 

3.29126 
3.29172 
3.29217 
3.29263 
3.29309 

40 
41 
42 

43 
44 

1611  .5 
1613  .3 

1615  .2 
1617  .1 
1619  »6 

3.20722 

3-20772 
3.20822 
3.20873 
3.20924 

1725  .6 
i727  -5 
'729  -5 
i73i  -5 
1733  '-4 

3.23697 
3-23745 
3-23794 
3-23843 
3.73801 

1843  -8 
1845  .8 
1847  .8 
1849  .8 
1251  8 

3-26571 
3.26619 
3.26666 
3.267.3 
3.26760 

1965  -8 
1967  .9 
1970  .0 
1972  .0 
1974  .1 

3-29354 
3.29400 
3.29446 
3.29491 

3.29537 

45 
46 

3 

49 

1620  .8 
1622  .7 
1624  .6  j 
1626  .5 
1628  .3 

3.20974 
3.21024 
*  21075 
21125 
3-2H-5 

J735  -3 

1737  -2 
1739  -2 
1741  .2 

J743  -1 

3.23940 
3.23988 
3-24037 
3.24086 
3-24134 

1853  .8 
1855  -8 
1857  -8 
1859  -8 
1861  .9 

3.26807 
3-26854 
3.26901 
3.26948 
3-26995 

1976  .2 
1978  .2 
I980  .3 
1982  .4 
1984  .5 

3-29582 
3  29628 
3-29673 
3.29719 
3.29764 

50 

Si 
52 
53 
54 

1630  .2 
1632  .1 

1634  .0 
'635  -9 
1637  -7 

3.21225 
3-21275 
3-2I325 
3-21375 
3.21425 

1745  -i 
1747  -o 
1749  -° 
1750  -9 
1752  .8 

3.24182 
3.24231 
3.24279 
3-24328 
3  24176 

1863  .9 
1865  .9 
1867  .9 
1869  .9 
1871  .9 

3.27042 
3.27088 
3-27135 
3.27182 
3.27229 

1986  .5 

1988  .6 
1990  .7 
1902  .8 
1994  .8 

3.29810 
3-29855 
3.29900 
3.29946 
3.29991 

55 
56 
57 
58 
59 

1639  .6 
1641  .5 
1643  .4 
I645  -3 

1647  .2 

3-21475 
3-2I525 
3-21575 
3-21625 
3.21675 

1754  -8 
1756  .8 
1758  -7 
1760  .7 
1762  .6 

3.24424 

3.24473 
3-24521 
3.24s69 
3.24617 

1873  -9 
1876  .0 
1878  .0 
1880  .0 
1882  .0 

3-27276 
3.27322 
3-27369 
3.27416 
3.27462 

1906  .9 
1999  .0 

2OOI  .1 
2003  .2 
2005  .3 

3.30036 
3.30082 
3.30127 
3.30172 

3.30217 

60 

1649  .1 

3-2I725 

1764  .6 

3  24665 

1884  .0 

3-27509 

2007  .4 

3.30262 

642 


TABLE    VIII  B. 


TABLE    VIII  C 


2  sin4  \t 
sin  i" 


L__-j9- 

~  86^00"  J 


t 

n 

log  « 

/ 

n 

log  w 

Om  0« 
I   O 

2   0 

3  o 
4  o 
5  o 

o".oo 
o  .00 

0  .00 
0  .00 

o  .00 

0  .01 

4.9706 
6.1747 
6.8791 

7'3^8 
7.7665 

22m  O8 
10 
2O 
30 
40 
50 

2".  I9 
2  .25 
2  .32 

2  '32 
2  .46 

2  -54 

0.3396 
0-3527 
0.3657 
0.3786 
°-39'5 
0.4042 

6  o 

7  o 
8  o 
9  o 

IO   O 
II   0 

0  .01 

o  .02 
o  .04 
o  .06 
o  .09 
o  .13 

8.0832 
8.3509 
8.5829 
8.7875 

8  •  97°5 
9.1360 

23  o 

IO 
20 

3° 
4° 
50 

2  .6l 

2  .69 
2  .77 
2  .85 
2  -93 

3  -oi 

0.4168 
o  4293 
0.4418 
0.4541 
0.4664 
0.4786 

12   0 

12  30 

13  o 

13  3° 
14  o 
14  30 

o  .19 
o  .23 
o  .27 
o  .31 
o  .36 
o  .41 

9.2871 
9.3580 
9.4262 
9.4917 
9  5549 
9.6158 

24  o 

IO 
20 

3° 
40 
50 

3  -io 
3  -18 
3  -27 

3  -36 
3  -4S 

3  -5=; 

0.4907 
0.5027 
0.5146 
0.5264 
0.5382 
0  5499 

15  o 

10 
20 

3° 
4o 
50 

o  .47 
o  .49 
o  .52 
o  .54 
o  .56 
o  -59 

9.6747 
9.6939 
9.7128 

9-73l6 
9.7502 
9  7686 

25  o 

IO 
20 
3° 
40 
50 

3  -('4 

:« 

•94 
•°5 
•15 

o  56l5 
o.573o 
0.5845 
0-5959 
0.6072 
o  6184 

16  o 

10 
20 
3« 
40 

50 

o  .61 
o  .64 
o  .67 
o  .69 
o  .72 
o  -75 

9.7867 
9.8047 
9.8225 
9.8402 
9.8576 
9.8749 

26  o 

IO 
20 

3° 
40 

5° 

.26 

| 

.60 

£ 

0.6296 
0.6407 
0.6517 
o  6626 
0-6735 
0.6843 

17  o 

IO 
20 
30 
40 

5° 

o  .78 
o  .81 
o  .84 
o  .88 
o  .91 
o  -95 

9  .  8920 
9.9089 
9-9257 
9.9423 
9.9588 
9-9751 

27  o 

IO 
20 

3° 
40 

5° 

4  .96 
5  -08 
5  -20 
5  -33 
5  -46 
5  -60 

0.6951 
0.7057 
0.7164 
0.7269 
0-7374 
o  7478 

18  o 

IO 
20 

3° 

40 
50 

o  .98 

I  .02 

I  .06 
I  .09 

;::i 

9-99*3 
0.0072 
0.0231 
0.0388 
0.0544 
o  .  0698 

28  o 

IO 
20 

3<> 
40 

5° 

5  -73 
5  -87 
6  .01 
6  .15 
6  .30 
6  .44 

0.7582 
0.7685 
0.7787 
0.7889 
0.7990 

o  .  8090 

19  o 

10 
20 

30 
40 

5° 

I  .22 

I  .26 

I  .30 

i  -35 

I  .40 

I  .44 

0.0851 
0.1003 
0.1153 
o  .  i  302 

0.1450 
0.1597 

29  o 

IO 
20 

3° 
40 

50 

6  -59 
6  -75 
6  .90 
7  .06 
7  .22 
7  -38 

o  8190 
0.8290 

0/8389 
0.8487 
0.8585 
0.8682 

20  o 

IO 
20 

30 
40 
50 

I  .49 

1  -54 

I  .60 

I  .65 
I  .70 
I  .76 

0.1742 
0.1886 

0.2029 

O  .  2  I  70 
O.23II 
0.2450 

3*  o 

IO 
20 
30 
40 
50 

7  -55 
7  -72 
7  .89 
8  .06 
8  .24 
8  .42 

0.8778 

0.8874 
0.8970 
o  9065 
0.9160 

0.9254 

21   O 
10 
2O 

3° 
40 
50 

I  .82 

I  .87 

i  -93 
i  .99 

2  .06 
2  .12 

0.2589 
0.2726 
0.2862 
0.2997 
0.3131 
0.3264 

31  o 

10 

20 

3° 
40 

50 

8  .60 
8  .79 
8  .98 
9  -'7 
9  -37 
9  -57 

0.9347 
0.9440 

0-9533 
0.9625 
0.9716 
0.9807 

22m  0. 

2  .19 

o  .  3396 

32">0« 

9  -77 

o  9898 

Rate. 

I 

log£. 

-  3°' 

9.999  6985 

29 

7085 

28 

7186 

27 

7286 

26 

7387 

25 

7487 

24 

7588 

23 

7688 

22 

7789 

21 

7889 

20 

7990  J 

*9 

8090  [ 

18 

8191 

17 

8291 

16 

8392 

15 

8492 

14 

8593 

13 

869, 

12 

8794 

II 

8894 

IO 

8995 

I 

9095 
9196 

7 

9296 

6 

9397 

5 

9497 

4 

9598 

3 

,9698 

2 

9799 

—   I 

9899 

0 

0.000  0000 

+   I 

oioi 

2 

0201 

3 

t»302 

4 

0402 

5 

0503 

6 

0603 

7 

0704 

8 

0804 

9 

0905 

10 

1005 

ii 

1106 

12 

1206 

*3 

J307 

14 

1407 

15 

1508 

16 

1608 

17 

1709 

18 

1809 

19 

1910 

20 

2OIO 

21 

211  I 

22 

2212 

23 

23I2 

24 

2412 

25 

2S  I  "? 

26 

2613 

27 

2714 

28 

2814 

29 

29'  5 

+  30 

o  ooo  3015 

UNIVERSITY  OF  CALIFORNIA  LIBRARY 
BERKELEY 


Return  to  desk  from  which  borrowed. 
This  book  is  DUE  on  the  last  date  stamped  below. 


ASTR 


-NDMY   LIBRARY 


LD  21-100m-ll,'49(B7146sl6)476 


M30910 


THE  UNIVERSITY  OF  CALIFORNIA  LIBRARY 


